2. Wave Equations

The temporal domain \([0,T]\) is represented by a finite number of mesh points Similarly, the spatial domain \([0,L]\) is replaced by a set of mesh points One may view the mesh as two-dimensional in the x,t plane, consisting of points \((x_{i},t_{n})\), with \(i=0,\ldots,N_{x}\) and \(n=0,\ldots,N_{t}\).

The solution \(u(x,t)\) is sought at the mesh points. We introduce the mesh function \(u_{i}^{n}\), which approximates the exact solution at the mesh point \((x_{i},t_{n})\) for \(i=0,\ldots,N_{x}\) and \(n=0,\ldots,N_{t}\). Using the finite difference method, we shall develop algebraic equations for computing the mesh function.

In the finite difference method, we relax the condition that (2.1) holds at all points in the space-time domain \((0,L)\times(0,T]\) to the requirement that the PDE is fulfilled at the interior mesh points only: for \(i=1,\ldots,N_{x}-1\) and \(n=1,\ldots,N_{t}-1\). For n = 0 we have the initial conditions \(u=I(x)\) and \(u_{t}=0\), and at the boundaries \(i=0,N_{x}\) we have the boundary condition u = 0.

The second-order derivatives can be replaced by central differences. The most widely used difference approximation of the second-order derivative is It is convenient to introduce the finite difference operator notation A similar approximation of the second-order derivative in the x direction reads

We assume that \(u^{n}_{i}\) and \(u^{n-1}_{i}\) are available for \(i=0,\ldots,N_{x}\). The only unknown quantity in (2.11) is therefore \(u^{n+1}_{i}\), which we now can solve for: We have here introduced the parameter known as the Courant number.

Given that \(u^{n-1}_{i}\) and \(u^{n}_{i}\) are known for \(i=0,\ldots,N_{x}\), we find new values at the next time level by applying the formula (2.14) for \(i=1,\ldots,N_{x}-1\). Figure 2.1 illustrates the points that are used to compute \(u^{3}_{2}\). For the boundary points, i = 0 and \(i=N_{x}\), we apply the boundary conditions \(u_{i}^{n+1}=0\).

Even though sound reasoning leads up to (2.14), there is still a minor challenge with it that needs to be resolved. Think of the very first computational step to be made. The scheme (2.14) is supposed to start at n = 1, which means that we compute u ² from u ¹ and u ⁰. Unfortunately, we do not know the value of u ¹, so how to proceed? A standard procedure in such cases is to apply (2.14) also for n = 0. This immediately seems strange, since it involves \(u^{-1}_{i}\), which is an undefined quantity outside the time mesh (and the time domain). However, we can use the initial condition (2.13) in combination with (2.14) when n = 0 to eliminate \(u^{-1}_{i}\) and arrive at a special formula for \(u_{i}^{1}\): Figure 2.2 illustrates how (2.16) connects four instead of five points: \(u^{1}_{2}\), \(u_{1}^{0}\), \(u_{2}^{0}\), and \(u_{3}^{0}\).

We can now summarize the computational algorithm:The algorithm essentially consists of moving a finite difference stencil through all the mesh points, which can be seen as an animation in a web page ¹ or a movie file ².

The algorithm only involves the three most recent time levels, so we need only three arrays for \(u_{i}^{n+1}\), \(u_{i}^{n}\), and \(u_{i}^{n-1}\), \(i=0,\ldots,N_{x}\). Storing all the solutions in a two-dimensional array of size \((N_{x}+1)\times(N_{t}+1)\) would be possible in this simple one-dimensional PDE problem, but is normally out of the question in three-dimensional (3D) and large two-dimensional (2D) problems. We shall therefore, in all our PDE solving programs, have the unknown in memory at as few time levels as possible.

In a Python implementation of this algorithm, we use the array elements u[i] to store \(u^{n+1}_{i}\), u_n[i] to store \(u^{n}_{i}\), and u_nm1[i] to store \(u^{n-1}_{i}\).

The following Python snippet realizes the steps in the computational algorithm.

We now address the following extended initial-boundary value problem for one-dimensional wave phenomena:

Sampling the PDE at \((x_{i},t_{n})\) and using the same finite difference approximations as above, yields Writing this out and solving for the unknown \(u^{n+1}_{i}\) results in

The equation for the first time step must be rederived. The discretization of the initial condition \(u_{t}=V(x)\) at t = 0 becomes which, when inserted in (2.23) for n = 0, gives the special formula

Many wave problems feature sinusoidal oscillations in time and space. For example, the original PDE problem (2.1)–(2.5) allows an exact solution This \(u_{\mbox{\footnotesize e}}\) fulfills the PDE with f = 0, boundary conditions \(u_{\mbox{\footnotesize e}}(0,t)=u_{\mbox{\footnotesize e}}(L,t)=0\), as well as initial conditions \(I(x)=A\sin\left(\frac{\pi}{L}x\right)\) and V = 0.

An introduction to the computing of convergence rates is given in Section 3.1.6 in [9]. There is also a detailed example on computing convergence rates in Sect. 1.​2.​2.

In the present problem, one expects the method to have a convergence rate of 2 (see Sect. 2.10), so if the computed rates are close to 2 on a sufficiently fine mesh, we have good evidence that the implementation is free of programming mistakes.

Inserted in the PDE \(u_{tt}=c^{2}u_{xx}+f\) we get The initial conditions become

By using an exact solution of the PDE problem, we will next compute the error measure E on a sequence of refined meshes and see if the rates r = p = 2 are obtained. We will not be concerned with estimating the constants C _t and C _x , simply because we are not interested in their values.

It is advantageous to introduce a single discretization parameter \(h=\Delta t=\hat{c}\Delta x\) for some constant \(\hat{c}\). Since \(\Delta t\) and \(\Delta x\) are related through the Courant number, \(\Delta t=C\Delta x/c\), we set \(h=\Delta t\), and then \(\Delta x=hc/C\). Now the expression for the error measure is greatly simplified:

An alternative error measure is to use a spatial norm at one time step only, e.g., the end time T (\(n=N_{t}\)): The important point is that the error measure (E) for the simulation is represented by a single number.

Since r depends on i, i.e., which simulations we compare, we add an index to r: r _i , where \(i=0,\ldots,m-2\), if we have m experiments: \((h_{0},E_{0}),\ldots,(h_{m-1},E_{m-1})\).

In our present discretization of the wave equation we expect r = 2, and hence the r _i values should converge to 2 as i increases.

With a manufactured or known analytical solution, as outlined above, we can estimate convergence rates and see if they have the correct asymptotic behavior. Experience shows that this is a quite good verification technique in that many common bugs will destroy the convergence rates. A significantly better test though, would be to check that the numerical solution is exactly what it should be. This will in general require exact knowledge of the numerical error, which we do not normally have (although we in Sect. 2.10 establish such knowledge in simple cases). However, it is possible to look for solutions where we can show that the numerical error vanishes, i.e., the solution of the original continuous PDE problem is also a solution of the discrete equations. This property often arises if the exact solution of the PDE is a lower-order polynomial. (Truncation error analysis leads to error measures that involve derivatives of the exact solution. In the present problem, the truncation error involves 4th-order derivatives of u in space and time. Choosing u as a polynomial of degree three or less will therefore lead to vanishing error.)

We shall now illustrate the construction of an exact solution to both the PDE itself and the discrete equations. Our chosen manufactured solution is quadratic in space and linear in time. More specifically, we set which by insertion in the PDE leads to \(f(x,t)=2(1+t)c^{2}\). This \(u_{\mbox{\footnotesize e}}\) fulfills the boundary conditions u = 0 and demands \(I(x)=x(L-x)\) and \(V(x)={\frac{1}{2}}x(L-x)\).

To realize that the chosen \(u_{\mbox{\footnotesize e}}\) is also an exact solution of the discrete equations, we first remind ourselves that \(t_{n}=n\Delta t\) so that Hence, Similarly, we get that Now, \(f^{n}_{i}=2(1+{\frac{1}{2}}t_{n})c^{2}\), which results in

Moreover, \(u_{\mbox{\footnotesize e}}(x_{i},0)=I(x_{i})\), \(\partial u_{\mbox{\footnotesize e}}/\partial t=V(x_{i})\) at t = 0, and \(u_{\mbox{\footnotesize e}}(x_{0},t)=u_{\mbox{\footnotesize e}}(x_{N_{x}},0)=0\). Also the modified scheme for the first time step is fulfilled by \(u_{\mbox{\footnotesize e}}(x_{i},t_{n})\).

Therefore, the exact solution \(u_{\mbox{\footnotesize e}}(x,t)=x(L-x)(1+t/2)\) of the PDE problem is also an exact solution of the discrete problem. This means that we know beforehand what numbers the numerical algorithm should produce. We can use this fact to check that the computed \(u^{n}_{i}\) values from an implementation equals \(u_{\mbox{\footnotesize e}}(x_{i},t_{n})\), within machine precision. This result is valid regardless of the mesh spacings \(\Delta x\) and \(\Delta t\)! Nevertheless, there might be stability restrictions on \(\Delta x\) and \(\Delta t\), so the test can only be run for a mesh that is compatible with the stability criterion (which in the present case is C ≤ 1, to be derived later).

The solution at all spatial points at a new time level is stored in an array u of length \(N_{x}+1\). We need to decide what to do with this solution, e.g., visualize the curve, analyze the values, or write the array to file for later use. The decision about what to do is left to the user in the form of a user-supplied function

where u is the solution at the spatial points x at time t[n]. The user_action function is called from the solver at each time level n.

is a callback function that will terminate the solver function (given below) of the amplitude of the waves exceed 10, which is here considered as a numerical instability.

A first attempt at a solver function is listed below.

A couple of remarks about the above code is perhaps necessary:

We use the test problem derived in Sect. 2.2.1 for verification. Below is a unit test based on this test problem and realized as a proper test function compatible with the unit test frameworks nose or pytest.

When this function resides in the file wave1D_u0.py, one can run pytest to check that all test functions with names test_*() in this file work:

A more general method, but not so reliable as a verification method, is to compute the convergence rates and see if they coincide with theoretical estimates. Here we expect a rate of 2 according to the various results in Sect. 2.10. A general function for computing convergence rates can be written like this:

Using the analytical solution from Sect. 2.2.2, we can call convergence_rates to see if we get a convergence rate that approaches 2 and use the final estimate of the rate in an assert statement such that this function becomes a proper test function:

Now that we have verified the implementation it is time to do a real computation where we also display evolution of the waves on the screen. Since the solver function knows nothing about what type of visualizations we may want, it calls the callback function user_action(u, x, t, n). We must therefore write this function and find the proper statements for plotting the solution.

A function inside another function, like plot_u_st in the above code segment, has access to and remembers all the local variables in the surrounding code inside the viz function (!). This is known in computer science as a closure and is very convenient to program with. For example, the plt and time modules defined outside plot_u are accessible for plot_u_st when the function is called (as user_action) in the solver function. Some may think, however, that a class instead of a closure is a cleaner and easier-to-understand implementation of the user action function, see Sect. 2.8.

The plot_u_st function just makes a standard SciTools plot command for plotting u as a function of x at time t[n]. To achieve a smooth animation, the plot command should take keyword arguments instead of being broken into separate calls to xlabel, ylabel, axis, time, and show. Several plot calls will automatically cause an animation on the screen. In addition, we want to save each frame in the animation to file. We then need a filename where the frame number is padded with zeros, here tmp_0000.png, tmp_0001.png, and so on. The proper printf construction is then tmp_%04d.png. Section 1.​3.​2 contains more basic information on making animations.

The solver is called with an argument plot_u as user_function. If the user chooses to use SciTools, plot_u is the plot_u_st callback function, but for Matplotlib it is an instance of the class PlotMatplotlib. Also this class makes use of variables defined in the viz function: plt and time. With Matplotlib, one has to make the first plot the standard way, and then update the y data in the plot at every time level. The update requires active use of the returned value from plt.plot in the first plot. This value would need to be stored in a local variable if we were to use a closure for the user_action function when doing the animation with Matplotlib. It is much easier to store the variable as a class attribute self.lines. Since the class is essentially a function, we implement the function as the special method __call__ such that the instance plot_u(u, x, t, n) can be called as a standard callback function from solver.

The viz function creates an ffmpeg or avconv command with the proper arguments for each of the formats Flash, MP4, WebM, and Ogg. The task is greatly simplified by having a codec2ext dictionary for mapping video codec names to filename extensions. As mentioned in Sect. 1.​3.​2, only two formats are actually needed to ensure that all browsers can successfully play the video: MP4 and WebM.

Some animations having a large number of plot files may not be properly combined into a video using ffmpeg or avconv. A method that always works is to play the PNG files as an animation in a browser using JavaScript code in an HTML file. The SciTools package has a function movie (or a stand-alone command scitools movie) for creating such an HTML player. The plt.movie call in the viz function shows how the function is used. The file movie.html can be loaded into a browser and features a user interface where the speed of the animation can be controlled. Note that the movie in this case consists of the movie.html file and all the frame files tmp_*.png.

A simple choice of numbers may illustrate the formulas: say we have 801 frames in total (t.size) and we allow only 60 frames to be plotted. As n then runs from 801 to 0, we need to plot every 801/60 frame, which with integer division yields 13 as skip_frame. Using the mod function, n % skip_frame, this operation is zero every time n can be divided by 13 without a remainder. That is, the if test is true when n equals \(0,13,26,39,{\ldots},780,801\). The associated code is included in the plot_u function, inside the viz function, in the file wave1D_u0.py .

The first demo of our 1D wave equation solver concerns vibrations of a string that is initially deformed to a triangular shape, like when picking a guitar string: We choose L = 75 cm, \(x_{0}=0.8L\), a = 5 mm, and a time frequency ν = 440 Hz. The relation between the wave speed c and ν is \(c=\nu\lambda\), where λ is the wavelength, taken as 2L because the longest wave on the string forms half a wavelength. There is no external force, so f = 0 (meaning we can neglect gravity), and the string is at rest initially, implying V = 0.

Regarding numerical parameters, we need to specify a \(\Delta t\). Sometimes it is more natural to think of a spatial resolution instead of a time step. A natural semi-coarse spatial resolution in the present problem is \(N_{x}=50\). We can then choose the associated \(\Delta t\) (as required by the viz and solver functions) as the stability limit: \(\Delta t=L/(N_{x}c)\). This is the \(\Delta t\) to be specified, but notice that if C < 1, the actual \(\Delta x\) computed in solver gets larger than \(L/N_{x}\): \(\Delta x=c\Delta t/C=L/(N_{x}C)\). (The reason is that we fix \(\Delta t\) and adjust \(\Delta x\), so if C gets smaller, the code implements this effect in terms of a larger \(\Delta x\).)

The associated program has the name wave1D_u0.py . Run the program and watch the movie of the vibrating string ³. The string should ideally consist of straight segments, but these are somewhat wavy due to numerical approximation. Run the case with the wave1D_u0.py code and C = 1 to see the exact solution.

Depending on the model, it may be a substantial job to establish consistent and relevant physical parameter values for a case. The guitar string example illustrates the point. However, by scaling the mathematical problem we can often reduce the need to estimate physical parameters dramatically. The scaling technique consists of introducing new independent and dependent variables, with the aim that the absolute values of these lie in \([0,1]\). We introduce the dimensionless variables (details are found in Section 3.1.1 in [11]) Here, L is a typical length scale, e.g., the length of the domain, and a is a typical size of u, e.g., determined from the initial condition: \(a=\max_{x}|I(x)|\).

We get by the chain rule that Similarly, Inserting the dimensionless variables in the PDE gives, in case f = 0, Dropping the bars, we arrive at the scaled PDE which has no parameter c ² anymore. The initial conditions are scaled as and resulting in In the common case V = 0 we see that there are no physical parameters to be estimated in the PDE model!

If we have a program implemented for the physical wave equation with dimensions, we can obtain the dimensionless, scaled version by setting c = 1. The initial condition of a guitar string, given in (2.33), gets its scaled form by choosing a = 1, L = 1, and \(x_{0}\in[0,1]\). This means that we only need to decide on the x ₀ value as a fraction of unity, because the scaled problem corresponds to setting all other parameters to unity. In the code we can just set a=c=L=1, x0=0.8, and there is no need to calculate with wavelengths and frequencies to estimate c!

The only non-trivial parameter to estimate in the scaled problem is the final end time of the simulation, or more precisely, how it relates to periods in periodic solutions in time, since we often want to express the end time as a certain number of periods. The period in the dimensionless problem is 2, so the end time can be set to the desired number of periods times 2.

Why the dimensionless period is 2 can be explained by the following reasoning. Suppose that u behaves as \(\cos(\omega t)\) in time in the original problem with dimensions. The corresponding period is then \(P=2\pi/\omega\), but we need to estimate ω. A typical solution of the wave equation is \(u(x,t)=A\cos(kx)\cos(\omega t)\), where A is an amplitude and k is related to the wave length λ in space: \(\lambda=2\pi/k\). Both λ and A will be given by the initial condition I(x). Inserting this \(u(x,t)\) in the PDE yields \(-\omega^{2}=-c^{2}k^{2}\), i.e., ω = kc. The period is therefore \(P=2\pi/(kc)\). If the boundary conditions are \(u(0,t)=u(L,t)\), we need to have \(kL=n\pi\) for integer n. The period becomes P = 2L ∕ nc. The longest period is P = 2L ∕ c. The dimensionless period \(\tilde{P}\) is obtained by dividing P by the time scale L ∕ c, which results in \(\tilde{P}=2\). Shorter waves in the initial condition will have a dimensionless shorter period \(\tilde{P}=2/n\) (n > 1).

Efficient computing with numpy arrays demands that we avoid loops and compute with entire arrays at once (or at least large portions of them). Consider this calculation of differences \(d_{i}=u_{i+1}-u_{i}\):

All the differences here are independent of each other. The computation of d can therefore alternatively be done by subtracting the array \((u_{0},u_{1},\ldots,u_{n-1})\) from the array where the elements are shifted one index upwards: \((u_{1},u_{2},\ldots,u_{n})\), see Fig. 2.3. The former subset of the array can be expressed by u[0:n-1], u[0:-1], or just u[:-1], meaning from index 0 up to, but not including, the last element (-1). The latter subset is obtained by u[1:n] or u[1:], meaning from index 1 and the rest of the array. The computation of d can now be done without an explicit Python loop:

or with explicit limits if desired:

Indices with a colon, going from an index to (but not including) another index are called slices. With numpy arrays, the computations are still done by loops, but in efficient, compiled, highly optimized C or Fortran code. Such loops are sometimes referred to as vectorized loops. Such loops can also easily be distributed among many processors on parallel computers. We say that the scalar code above, working on an element (a scalar) at a time, has been replaced by an equivalent vectorized code. The process of vectorizing code is called vectorization.

Finite difference schemes basically contain differences between array elements with shifted indices. As an example, consider the updating formula

Note that the length of u2 becomes n-2. If u2 is already an array of length n and we want to use the formula to update all the ‘‘inner’’ elements of u2, as we will when solving a 1D wave equation, we can write

We need three temporary arrays, but a user does not need to worry about such temporary arrays.

Vectorization may also work nicely with functions. To illustrate, we may extend the previous example as follows:

Assuming u2, u, and x all have length n, the vectorized version becomes

Obviously, f must be able to take an array as argument for f(x[1:-1]) to make sense.

We now have the necessary tools to vectorize the wave equation algorithm as described mathematically in Sect. 2.1.5 and through code in Sect. 2.3.2. There are three loops: one for the initial condition, one for the first time step, and finally the loop that is repeated for all subsequent time levels. Since only the latter is repeated a potentially large number of times, we limit our vectorization efforts to this loop. Within the time loop, the space loop reads:

The vectorized version becomes

or

The program wave1D_u0v.py contains a new version of the function solver where both the scalar and the vectorized loops are included (the argument version is set to scalar or vectorized, respectively).

We may reuse the quadratic solution \(u_{\mbox{\footnotesize e}}(x,t)=x(L-x)(1+{\frac{1}{2}}t)\) for verifying also the vectorized code. A test function can now verify both the scalar and the vectorized version. Moreover, we may use a user_action function that compares the computed and exact solution at each time level and performs a test:

The wave1D_u0v.py contains our new solver function with both scalar and vectorized code. For comparing the efficiency of scalar versus vectorized code, we need a viz function as discussed in Sect. 2.3.5. All of this viz function can be reused, except the call to solver_function. This call lacks the parameter version, which we want to set to vectorized and scalar for our efficiency measurements.

One solution is to copy the viz code from wave1D_u0 into wave1D_u0v.py and add a version argument to the solver_function call. Taking into account how much animation code we then duplicate, this is not a good idea. Alternatively, introducing the version argument in wave1D_u0.viz, so that this function can be imported into wave1D_u0v.py, is not a good solution either, since version has no meaning in that file. We need better ideas!

The new scalar_solver takes the same arguments as wave1D_u0.scalar and calls wave1D_u0v.scalar, but always supplies the extra argument version= ’scalar’. When sending this solver_function to wave1D_u0.viz, the latter will call wave1D_u0v.solver with all the I, V, f, etc., arguments we supply, plus version=’scalar’.

At the end of each time step we need to update the u_nm1 and u_n arrays such that they have the right content for the next time step:

The order here is important: updating u_n first, makes u_nm1 equal to u, which is wrong!

The assignment u_n[:] = u copies the content of the u array into the elements of the u_n array. Such copying takes time, but that time is negligible compared to the time needed for computing u from the finite difference formula, even when the formula has a vectorized implementation. However, efficiency of program code is a key topic when solving PDEs numerically (particularly when there are two or three space dimensions), so it must be mentioned that there exists a much more efficient way of making the arrays u_nm1 and u_n ready for the next time step. The idea is based on switching references and explained as follows.

A Python variable is actually a reference to some object (C programmers may think of pointers). Instead of copying data, we can let u_nm1 refer to the u_n object and u_n refer to the u object. This is a very efficient operation (like switching pointers in C). A naive implementation like

will fail, however, because now u_nm1 refers to the u_n object, but then the name u_n refers to u, so that this u object has two references, u_n and u, while our third array, originally referred to by u_nm1, has no more references and is lost. This means that the variables u, u_n, and u_nm1 refer to two arrays and not three. Consequently, the computations at the next time level will be messed up, since updating the elements in u will imply updating the elements in u_n too, thereby destroying the solution at the previous time step.

While u_nm1 = u_n is fine, u_n = u is problematic, so the solution to this problem is to ensure that u points to the u_nm1 array. This is mathematically wrong, but new correct values will be filled into u at the next time step and make it right.

The correct switch of references is

This switching of references for updating our arrays will be used in later implementations.

When a wave hits a boundary and is to be reflected back, one applies the condition The derivative \(\partial/\partial n\) is in the outward normal direction from a general boundary. For a 1D domain \([0,L]\), we have that

How can we incorporate the condition (2.35) in the finite difference scheme? Since we have used central differences in all the other approximations to derivatives in the scheme, it is tempting to implement (2.35) at x = 0 and \(t=t_{n}\) by the difference The problem is that \(u_{-1}^{n}\) is not a u value that is being computed since the point is outside the mesh. However, if we combine (2.36) with the scheme for i = 0, we can eliminate the fictitious value \(u_{-1}^{n}\). We see that \(u_{-1}^{n}=u_{1}^{n}\) from (2.36), which can be used in (2.37) to arrive at a modified scheme for the boundary point \(u_{0}^{n+1}\): Figure 2.4 visualizes this equation for computing \(u^{3}_{0}\) in terms of \(u^{2}_{0}\), \(u^{1}_{0}\), and \(u^{2}_{1}\).

Similarly, (2.35) applied at x = L is discretized by a central difference Combined with the scheme for \(i=N_{x}\) we get a modified scheme for the boundary value \(u_{N_{x}}^{n+1}\):

The modification of the scheme at the boundary is also required for the special formula for the first time step. How the stencil moves through the mesh and is modified at the boundary can be illustrated by an animation in a web page ⁹ or a movie file ¹⁰.

We have seen in the preceding section that the special formulas for the boundary points arise from replacing \(u_{i-1}^{n}\) by \(u_{i+1}^{n}\) when computing \(u_{i}^{n+1}\) from the stencil formula for i = 0. Similarly, we replace \(u_{i+1}^{n}\) by \(u_{i-1}^{n}\) in the stencil formula for \(i=N_{x}\). This observation can conveniently be used in the coding: we just work with the general stencil formula, but write the code such that it is easy to replace u[i-1] by u[i+1] and vice versa. This is achieved by having the indices i+1 and i-1 as variables ip1 (i plus 1) and im1 (i minus 1), respectively. At the boundary we can easily define im1=i+1 while we use im1=i-1 in the internal parts of the mesh. Here are the details of the implementation (note that the updating formula for u[i] is the general stencil formula):

We can in fact create one loop over both the internal and boundary points and use only one updating formula:

The program wave1D_n0.py contains a complete implementation of the 1D wave equation with boundary conditions \(u_{x}=0\) at x = 0 and x = L.

It would be nice to modify the test_quadratic test case from the wave1D_u0.py with Dirichlet conditions, described in Sect. 2.4.3. However, the Neumann conditions require the polynomial variation in the x direction to be of third degree, which causes challenging problems when designing a test where the numerical solution is known exactly. Exercise 2.15 outlines ideas and code for this purpose. The only test in wave1D_n0.py is to start with a plug wave at rest and see that the initial condition is reached again perfectly after one period of motion, but such a test requires C = 1 (so the numerical solution coincides with the exact solution of the PDE, see Sect. 2.10.4).

To improve our mathematical writing and our implementations, it is wise to introduce a special notation for index sets. This means that we write x _i , followed by \(i\in\mathcal{I}_{x}\), instead of \(i=0,\ldots,N_{x}\). Obviously, \(\mathcal{I}_{x}\) must be the index set \(\mathcal{I}_{x}=\{0,\ldots,N_{x}\}\), but it is often advantageous to have a symbol for this set rather than specifying all its elements (all the time, as we have done up to now). This new notation saves writing and makes specifications of algorithms and their implementation as computer code simpler.

The first index in the set will be denoted \(\mathcal{I}_{x}^{0}\) and the last \(\mathcal{I}_{x}^{-1}\). When we need to skip the first element of the set, we use \(\mathcal{I}_{x}^{+}\) for the remaining subset \(\mathcal{I}_{x}^{+}=\{1,\ldots,N_{x}\}\). Similarly, if the last element is to be dropped, we write \(\mathcal{I}_{x}^{-}=\{0,\ldots,N_{x}-1\}\) for the remaining indices. All the indices corresponding to inner grid points are specified by \(\mathcal{I}_{x}^{i}=\{1,\ldots,N_{x}-1\}\). For the time domain we find it natural to explicitly use 0 as the first index, so we will usually write n = 0 and t ₀ rather than \(n=\mathcal{I}_{t}^{0}\). We also avoid notation like \(x_{\mathcal{I}_{x}^{-1}}\) and will instead use x _i , \(i=\mathcal{I}_{x}^{-1}\).

The Python code associated with index sets applies the following conventions:

For the current problem setting in the x,t plane, we work with the index sets defined in Python as

A finite difference scheme can with the index set notation be specified as The corresponding implementation becomes

How can we test that the Neumann conditions are correctly implemented? The solver function in the wave1D_dn.py program described in the box above accepts Dirichlet or Neumann conditions at x = 0 and x = L. It is tempting to apply a quadratic solution as described in Sect. 2.2.1 and 2.3.3, but it turns out that this solution is no longer an exact solution of the discrete equations if a Neumann condition is implemented on the boundary. A linear solution does not help since we only have homogeneous Neumann conditions in wave1D_dn.py, and we are consequently left with testing just a constant solution: \(u=\hbox{const}\).

The quadratic solution is very useful for testing, but it requires Dirichlet conditions at both ends.

Another test may utilize the fact that the approximation error vanishes when the Courant number is unity. We can, for example, start with a plug profile as initial condition, let this wave split into two plug waves, one in each direction, and check that the two plug waves come back and form the initial condition again after ‘‘one period’’ of the solution process. Neumann conditions can be applied at both ends. A proper test function reads

Other tests must rely on an unknown approximation error, so effectively we are left with tests on the convergence rate.

The important idea is to ensure that we always have because then the application of the standard scheme at a boundary point i = 0 or \(i=N_{x}\) will be correct and guarantee that the solution is compatible with the boundary condition \(u_{x}=0\).

Some readers may find it strange to just extend the domain with ghost cells as a general technique, because in some problems there is a completely different medium with different physics and equations right outside of a boundary. Nevertheless, one should view the ghost cell technique as a purely mathematical technique, which is valid in the limit \(\Delta x\rightarrow 0\) and helps us to implement derivatives.

Unfortunately, a major indexing problem arises with ghost cells. The reason is that Python indices must start at 0 and u[-1] will always mean the last element in u. This fact gives, apparently, a mismatch between the mathematical indices \(i=-1,0,\ldots,N_{x}+1\) and the Python indices running over u: 0,..,Nx+2. One remedy is to change the mathematical indexing of i in the scheme and write instead of \(i=0,\ldots,N_{x}\) as we have previously used. The ghost points now correspond to i = 0 and \(i=N_{x}+1\). A better solution is to use the ideas of Sect. 2.6.4: we hide the specific index value in an index set and operate with inner and boundary points using the index set notation.

To this end, we define u with proper length and Ix to be the corresponding indices for the real physical mesh points (\(1,2,\ldots,N_{x}+1\)):

That is, the boundary points have indices Ix[0] and Ix[-1] (as before). We first update the solution at all physical mesh points (i.e., interior points in the mesh):

The indexing becomes a bit more complicated when we call functions like V(x) and f(x, t), as we must remember that the appropriate x coordinate is given as x[i-Ix[0]]:

It remains to update the solution at ghost points, i.e., u[0] and u[-1] (or u[Nx+2]). For a boundary condition \(u_{x}=0\), the ghost value must equal the value at the associated inner mesh point. Computer code makes this statement precise:

The physical solution to be plotted is now in u[1:-1], or equivalently u[Ix[0]: Ix[-1]+1], so this slice is the quantity to be returned from a solver function. A complete implementation appears in the program wave1D_n0_ghost.py .

The ghost cell is only added to the boundary where we have a Neumann condition. Suppose we have a Dirichlet condition at x = L and a homogeneous Neumann condition at x = 0. One ghost cell \([-\Delta x,0]\) is added to the mesh, so the index set for the physical points becomes \(\{1,\ldots,N_{x}+1\}\). A relevant implementation is

The physical solution to be plotted is now in u[1:] or (as always) u[Ix[0]: Ix[-1]+1].

Instead of working with the squared quantity \(c^{2}(x)\), we shall for notational convenience introduce \(q(x)=c^{2}(x)\). A 1D wave equation with variable wave velocity often takes the form This is the most frequent form of a wave equation with variable wave velocity, but other forms also appear, see Sect. 2.14.1 and equation (2.125).

As usual, we sample (2.42) at a mesh point, where the only new term to discretize is

The principal idea is to first discretize the outer derivative. Define and use a centered derivative around \(x=x_{i}\) for the derivative of ϕ: Then discretize Similarly, These intermediate results are now combined to With operator notation we can write the discretization as

If q is a known function of x, we can easily evaluate \(q_{i+\frac{1}{2}}\) simply as \(q(x_{i+\frac{1}{2}})\) with \(x_{i+\frac{1}{2}}=x_{i}+\frac{1}{2}\Delta x\). However, in many cases c, and hence q, is only known as a discrete function, often at the mesh points x _i . Evaluating q between two mesh points x _i and \(x_{i+1}\) must then be done by interpolation techniques, of which three are of particular interest in this context: The arithmetic mean in (2.46) is by far the most commonly used averaging technique and is well suited for smooth q(x) functions. The harmonic mean is often preferred when q(x) exhibits large jumps (which is typical for geological media). The geometric mean is less used, but popular in discretizations to linearize quadratic nonlinearities (see Sect. 1.​10.​2 for an example).

With the operator notation from (2.46) we can specify the discretization of the complete variable-coefficient wave equation in a compact way: Strictly speaking, \([D_{x}\overline{q}^{x}D_{x}u]^{n}_{i}=[D_{x}(\overline{q}^{x}D_{x}u)]^{n}_{i}\).

From the compact difference notation we immediately see what kind of differences that each term is approximated with. The notation \(\overline{q}^{x}\) also specifies that the variable coefficient is approximated by an arithmetic mean, the definition being \([\overline{q}^{x}]_{i+\frac{1}{2}}=(q_{i}+q_{i+1})/2\).

Before implementing, it remains to solve (2.49) with respect to \(u_{i}^{n+1}\):

The stability criterion derived later (Sect. 2.10.3) reads \(\Delta t\leq\Delta x/c\). If \(c=c(x)\), the criterion will depend on the spatial location. We must therefore choose a \(\Delta t\) that is small enough such that no mesh cell has \(\Delta t> \Delta x/c(x)\). That is, we must use the largest c value in the criterion: The parameter β is included as a safety factor: in some problems with a significantly varying c it turns out that one must choose β < 1 to have stable solutions (β = 0.9 may act as an all-round value).

A different strategy to handle the stability criterion with variable wave velocity is to use a spatially varying \(\Delta t\). While the idea is mathematically attractive at first sight, the implementation quickly becomes very complicated, so we stick to a constant \(\Delta t\) and a worst case value of c(x) (with a safety factor β).

Consider a Neumann condition \(\partial u/\partial x=0\) at \(x=L=N_{x}\Delta x\), discretized as for \(i=N_{x}\). Using the scheme (2.50) at the end point \(i=N_{x}\) with \(u_{i+1}^{n}=u_{i-1}^{n}\) results in Here we used the approximation

An alternative derivation may apply the arithmetic mean of \(q_{n-\frac{1}{2}}\) and \(q_{n+\frac{1}{2}}\) in (2.53), leading to the term Since \(\frac{1}{2}(q_{i+1}+q_{i-1})=q_{i}+{\cal O}(\Delta x^{2})\), we can approximate with \(2q_{i}(u_{i-1}^{n}-u_{i}^{n})\) for \(i=N_{x}\) and get the same term as we did above.

A common technique when implementing \(\partial u/\partial x=0\) boundary conditions, is to assume dq ∕ dx = 0 as well. This implies \(q_{i+1}=q_{i-1}\) and \(q_{i+1/2}=q_{i-1/2}\) for \(i=N_{x}\). The implications for the scheme are

The implementation of the scheme with a variable wave velocity \(q(x)=c^{2}(x)\) may assume that q is available as an array q[i] at the spatial mesh points. The following loop is a straightforward implementation of the scheme (2.50):

The coefficient C2 is now defined as (dt/dx)**2, i.e., not as the squared Courant number, since the wave velocity is variable and appears inside the parenthesis.

With Neumann conditions \(u_{x}=0\) at the boundary, we need to combine this scheme with the discrete version of the boundary condition, as shown in Sect. 2.7.5. Nevertheless, it would be convenient to reuse the formula for the interior points and just modify the indices ip1=i+1 and im1=i-1 as we did in Sect. 2.6.3. Assuming dq ∕ dx = 0 at the boundaries, we can implement the scheme at the boundary with the following code.

With ghost cells we can just reuse the formula for the interior points also at the boundary, provided that the ghost values of both u and q are correctly updated to ensure \(u_{x}=0\) and \(q_{x}=0\).

A vectorized version of the scheme with a variable coefficient at internal mesh points becomes

Sometimes a wave PDE has a variable coefficient in front of the time-derivative term: One example appears when modeling elastic waves in a rod with varying density, cf. (2.14.1) with \(\varrho(x)\).

A natural scheme for (2.58) is We realize that the \(\varrho\) coefficient poses no particular difficulty, since \(\varrho\) enters the formula just as a simple factor in front of a derivative. There is hence no need for any averaging of \(\varrho\). Often, \(\varrho\) will be moved to the right-hand side, also without any difficulty:

Waves die out by two mechanisms. In 2D and 3D the energy of the wave spreads out in space, and energy conservation then requires the amplitude to decrease. This effect is not present in 1D. Damping is another cause of amplitude reduction. For example, the vibrations of a string die out because of damping due to air resistance and non-elastic effects in the string.

The simplest way of including damping is to add a first-order derivative to the equation (in the same way as friction forces enter a vibrating mechanical system): where b ≥ 0 is a prescribed damping coefficient.

A typical discretization of (2.61) in terms of centered differences reads Writing out the equation and solving for the unknown \(u^{n+1}_{i}\) gives the scheme for \(i\in\mathcal{I}_{x}^{i}\) and n ≥ 1. New equations must be derived for \(u^{1}_{i}\), and for boundary points in case of Neumann conditions.

The damping is very small in many wave phenomena and thus only evident for very long time simulations. This makes the standard wave equation without damping relevant for a lot of applications.

A useful feature in the wave1D_dn_vc.py program is the specification of the user_action function as a class. This part of the program may need some motivation and explanation. Although the plot_u_st function (and the PlotMatplotlib class) in the wave1D_u0.viz function remembers the local variables in the viz function, it is a cleaner solution to store the needed variables together with the function, which is exactly what a class offers.

The constructor shows how we can flexibly import the plotting engine as (typically) scitools.easyviz.gnuplot_ or scitools.easyviz.matplotlib_ (note the trailing underscore - it is required). With the screen_movie parameter we can suppress displaying each movie frame on the screen. Alternatively, for slow movies associated with fine meshes, one can set skip_frame=10, causing every 10 frames to be shown.

The __call__ method makes PlotAndStoreSolution instances behave like functions, so we can just pass an instance, say p, as the user_action argument in the solver function, and any call to user_action will be a call to p.__call__. The __call__ method plots the solution on the screen, saves the plot to file, and stores the solution in a file for later retrieval.

More details on storing the solution in files appear in Sect. C.2.

The function pulse in wave1D_dn_vc.py demonstrates wave motion in heterogeneous media where c varies. One can specify an interval where the wave velocity is decreased by a factor slowness_factor (or increased by making this factor less than one). Figure 2.5 shows a typical simulation scenario.

Four types of initial conditions are available:These peak-shaped initial conditions can be placed in the middle (loc=’center’) or at the left end (loc=’left’) of the domain. With the pulse in the middle, it splits in two parts, each with half the initial amplitude, traveling in opposite directions. With the pulse at the left end, centered at x = 0, and using the symmetry condition \(\partial u/\partial x=0\), only a right-going pulse is generated. There is also a left-going pulse, but it travels from x = 0 in negative x direction and is not visible in the domain \([0,L]\).

The PlotMediumAndSolution class used here is a subclass of PlotAndStore Solution where the medium with reduced c value, as specified by the medium interval, is visualized in the plots.

To easily kill the graphics by Ctrl-C and restart a new simulation it might be easier to run the above two statements from the command line with

The wave equation has solutions of the form for any functions g _R and g _L sufficiently smooth to be differentiated twice. The result follows from inserting (2.75) in the wave equation. A function of the form \(g_{R}(x-ct)\) represents a signal moving to the right in time with constant velocity c. This feature can be explained as follows. At time t = 0 the signal looks like \(g_{R}(x)\). Introducing a moving horizontal coordinate ξ = x − ct, we see the function \(g_{R}(\xi)\) is ‘‘at rest’’ in the ξ coordinate system, and the shape is always the same. Say the \(g_{R}(\xi)\) function has a peak at ξ = 0. This peak is located at x = ct, which means that it moves with the velocity dx ∕ dt = c in the x coordinate system. Similarly, \(g_{L}(x+ct)\) is a function, initially with shape \(g_{L}(x)\), that moves in the negative x direction with constant velocity c (introduce ξ = x + ct, look at the point ξ = 0, x = −ct, which has velocity dx ∕ dt = −c).

With the particular initial conditions we get, with u as in (2.75), The former suggests \(g_{R}=g_{L}\), and the former then leads to \(g_{R}=g_{L}=I/2\). Consequently, The interpretation of (2.76) is that the initial shape of u is split into two parts, each with the same shape as I but half of the initial amplitude. One part is traveling to the left and the other one to the right.

The solution has two important physical features: constant amplitude of the left and right wave, and constant velocity of these two waves. It turns out that the numerical solution will also preserve the constant amplitude, but the velocity depends on the mesh parameters \(\Delta t\) and \(\Delta x\).

The solution (2.76) will be influenced by boundary conditions when the parts \(\frac{1}{2}I(x-ct)\) and \(\frac{1}{2}I(x+ct)\) hit the boundaries and get, e.g., reflected back into the domain. However, when I(x) is nonzero only in a small part in the middle of the spatial domain \([0,L]\), which means that the boundaries are placed far away from the initial disturbance of u, the solution (2.76) is very clearly observed in a simulation.

A useful representation of solutions of wave equations is a linear combination of sine and/or cosine waves. Such a sum of waves is a solution if the governing PDE is linear and each sine or cosine wave fulfills the equation. To ease analytical calculations by hand we shall work with complex exponential functions instead of real-valued sine or cosine functions. The real part of complex expressions will typically be taken as the physical relevant quantity (whenever a physical relevant quantity is strictly needed). The idea now is to build I(x) of complex wave components e ^ikx : Here, k is the frequency of a component, K is some set of all the discrete k values needed to approximate I(x) well, and b _k are constants that must be determined. We will very seldom need to compute the b _k coefficients: most of the insight we look for, and the understanding of the numerical methods we want to establish, come from investigating how the PDE and the scheme treat a single component e ^ikx wave.

Letting the number of k values in K tend to infinity, makes the sum (2.77) converge to I(x). This sum is known as a Fourier series representation of I(x). Looking at (2.76), we see that the solution \(u(x,t)\), when I(x) is represented as in (2.77), is also built of basic complex exponential wave components of the form \(e^{ik(x\pm ct)}\) according to It is common to introduce the frequency in time ω = kc and assume that \(u(x,t)\) is a sum of basic wave components written as \(e^{ikx-\omega t}\). (Observe that inserting such a wave component in the governing PDE reveals that \(\omega^{2}=k^{2}c^{2}\), or ω = ±kc, reflecting the two solutions: one (+kc) traveling to the right and the other (−kc) traveling to the left.)

The above introduction to function representation by sine and cosine waves was quick and intuitive, but will suffice as background knowledge for the following material of single wave component analysis. However, to understand all details of how different wave components sum up to the analytical and numerical solutions, a more precise mathematical treatment is helpful and therefore summarized below.

It is well known that periodic functions can be represented by Fourier series. A generalization of the Fourier series idea to non-periodic functions defined on the real line is the Fourier transform: The function A(k) reflects the weight of each wave component e ^ikx in an infinite sum of such wave components. That is, A(k) reflects the frequency content in the function I(x). Fourier transforms are particularly fundamental for analyzing and understanding time-varying signals.

The solution of the linear 1D wave PDE can be expressed as

In a finite difference method, we represent u by a mesh function \(u^{n}_{q}\), where n counts temporal mesh points and q counts the spatial ones (the usual counter for spatial points, i, is here already used as imaginary unit). Similarly, I(x) is approximated by the mesh function I _q , \(q=0,\ldots,N_{x}\). On a mesh, it does not make sense to work with wave components e ^ikx for very large k, because the shortest possible sine or cosine wave that can be represented uniquely on a mesh with spacing \(\Delta x\) is the wave with wavelength \(2\Delta x\). This wave has its peaks and throughs at every two mesh points. That is, the wave ‘‘jumps up and down’’ between the mesh points.

The corresponding k value for the shortest possible wave in the mesh is \(k=2\pi/(2\Delta x)=\pi/\Delta x\). This maximum frequency is known as the Nyquist frequency. Within the range of relevant frequencies \((0,\pi/\Delta x]\) one defines the discrete Fourier transform ¹¹, using \(N_{x}+1\) discrete frequencies: The A _k values represent the discrete Fourier transform of the I _q values, which themselves are the inverse discrete Fourier transform of the A _k values.

The discrete Fourier transform is efficiently computed by the Fast Fourier transform algorithm. For a real function I(x), the relevant Python code for computing and plotting the discrete Fourier transform appears in the example below.

The scheme for the wave equation \(u_{tt}=c^{2}u_{xx}\) allows basic wave components as solution, but it turns out that the frequency in time, \(\tilde{\omega}\), is not equal to the exact frequency ω = kc. The goal now is to find exactly what \(\tilde{\omega}\) is. We ask two key questions:The following analysis will answer these questions. We shall continue using q as an identifier for a certain mesh point in the x direction.

Consider a right-hand side in (2.88) of magnitude larger than unity. The solution \(\tilde{\omega}\) of (2.88) must then be a complex number \(\tilde{\omega}=\tilde{\omega}_{r}+i\tilde{\omega}_{i}\) because the sine function is larger than unity for a complex argument. One can show that for any ω _i there will also be a corresponding solution with −ω _i . The component with \(\omega_{i}> 0\) gives an amplification factor \(e^{\omega_{i}t}\) that grows exponentially in time. We cannot allow this and must therefore require C ≤ 1 as a stability criterion.

Equation (2.88) can be solved with respect to \(\tilde{\omega}\): The relation between the numerical frequency \(\tilde{\omega}\) and the other parameters k, c, \(\Delta x\), and \(\Delta t\) is called a numerical dispersion relation. Correspondingly, ω = kc is the analytical dispersion relation. In general, dispersion refers to the phenomenon where the wave velocity depends on the spatial frequency (k, or the wave length \(\lambda=2\pi/k\)) of the wave. Since the wave velocity is ω ∕ k = c, we realize that the analytical dispersion relation reflects the fact that there is no dispersion. However, in a numerical scheme we have dispersive waves where the wave velocity depends on k.

The special case C = 1 deserves attention since then the right-hand side of (2.90) reduces to That is, \(\tilde{\omega}=\omega\) and the numerical solution is exact at all mesh points regardless of \(\Delta x\) and \(\Delta t\)! This implies that the numerical solution method is also an analytical solution method, at least for computing u at discrete points (the numerical method says nothing about the variation of u between the mesh points, and employing the common linear interpolation for extending the discrete solution gives a curve that in general deviates from the exact one).

For a closer examination of the error in the numerical dispersion relation when C < 1, we can study \(\tilde{\omega}-\omega\), \(\tilde{\omega}/\omega\), or the similar error measures in wave velocity: \(\tilde{c}-c\) and \(\tilde{c}/c\), where c = ω ∕ k and \(\tilde{c}=\tilde{\omega}/k\). It appears that the most convenient expression to work with is \(\tilde{c}/c\), since it can be written as a function of just two parameters: with \(p=k\Delta x/2\) as a non-dimensional measure of the spatial frequency. In essence, p tells how many spatial mesh points we have per wave length in space for the wave component with frequency k (recall that the wave length is \(2\pi/k\)). That is, p reflects how well the spatial variation of the wave component is resolved in the mesh. Wave components with wave length less than \(2\Delta x\) (\(2\pi/k<2\Delta x\)) are not visible in the mesh, so it does not make sense to have p > π ∕ 2.

We may introduce the function \(r(C,p)=\tilde{c}/c\) for further investigation of numerical errors in the wave velocity: This function is very well suited for plotting since it combines several parameters in the problem into a dependence on two dimensionless numbers, C and p.

Defining

we can plot \(r(C,p)\) as a function of p for various values of C, see Fig. 2.6. Note that the shortest waves have the most erroneous velocity, and that short waves move more slowly than they should.

We can also easily make a Taylor series expansion in the discretization parameter p:

Note that without the .removeO() call the series gets an O(x**7) term that makes it impossible to convert the series to a Python function (for, e.g., plotting).

From the rs_error_leading_order expression above, we see that the leading order term in the error of this series expansion is pointing to an error \(\mathcal{O}(\Delta t^{2},\Delta x^{2})\), which is compatible with the errors in the difference approximations (\(D_{t}D_{t}u\) and \(D_{x}D_{x}u\)).

We see from the last expression that C = 1 makes all the terms in rs vanish. Since we already know that the numerical solution is exact for C = 1, the remaining terms in the Taylor series expansion will also contain factors of C − 1 and cancel for C = 1.

The typical analytical solution of a 2D wave equation is a wave traveling in the direction of \(\boldsymbol{k}=k_{x}\boldsymbol{i}+k_{y}\boldsymbol{j}\), where i and j are unit vectors in the x and y directions, respectively (i should not be confused with \(i=\sqrt{-1}\) here). Such a wave can be expressed by for some twice differentiable function g, or with ω = kc, \(k=|\boldsymbol{k}|\): We can, in particular, build a solution by adding complex Fourier components of the form

A discrete 2D wave equation can be written as This equation admits a Fourier component as solution. Letting the operators \(D_{t}D_{t}\), \(D_{x}D_{x}\), and \(D_{y}D_{y}\) act on \(u^{n}_{q,r}\) from (2.94) transforms (2.93) to or where we have eliminated the factor 4 and introduced the symbols For a real-valued \(\tilde{\omega}\) the right-hand side must be less than or equal to unity in absolute value, requiring in general that This gives the stability criterion, more commonly expressed directly in an inequality for the time step: A similar, straightforward analysis for the 3D case leads to In the case of a variable coefficient \(c^{2}=c^{2}(\boldsymbol{x})\), we must use the worst-case value in the stability criteria. Often, especially in the variable wave velocity case, it is wise to introduce a safety factor \(\beta\in(0,1]\) too:

The exact numerical dispersion relations in 2D and 3D becomes, for constant c,

We can visualize the numerical dispersion error in 2D much like we did in 1D. To this end, we need to reduce the number of parameters in \(\tilde{\omega}\). The direction of the wave is parameterized by the polar angle θ, which means that A simplification is to set \(\Delta x=\Delta y=h\). Then \(C_{x}=C_{y}=c\Delta t/h\), which we call C. Also, The numerical frequency \(\tilde{\omega}\) is now a function of three parameters:We want to visualize the error in the numerical frequency. To avoid having \(\Delta t\) as a free parameter in \(\tilde{\omega}\), we work with \(\tilde{c}/c=\tilde{\omega}/(kc)\). The coefficient in front of the \(\sin^{-1}\) factor is then and We want to visualize this quantity as a function of p and θ for some values of C ≤ 1. It is instructive to make color contour plots of \(1-\tilde{c}/c\) in polar coordinates with θ as the angular coordinate and p as the radial coordinate.

The stability criterion (2.97) becomes \(C\leq C_{\max}=1/\sqrt{2}\) in the present 2D case with the C defined above. Let us plot \(1-\tilde{c}/c\) in polar coordinates for \(C_{\max},0.9C_{\max},0.5C_{\max},0.2C_{\max}\). The program below does the somewhat tricky work in Matplotlib, and the result appears in Fig. 2.7. From the figure we clearly see that the maximum C value gives the best results, and that waves whose propagation direction makes an angle of 45 degrees with an axis are the most accurate.

The general wave equation in d space dimensions, with constant wave velocity c, can be written in the compact form where in a 2D problem (d = 2) and in three space dimensions (d = 3).

Many applications involve variable coefficients, and the general wave equation in d dimensions is in this case written as which in, e.g., 2D becomes To save some writing and space we may use the index notation, where subscript t, x, or y means differentiation with respect to that coordinate. For example, These comments extend straightforwardly to 3D, which means that the 3D versions of the two wave PDEs, with and without variable coefficients, can be stated as

At each point of the boundary \(\partial\Omega\) (of Ω) we need one boundary condition involving the unknown u. The boundary conditions are of three principal types:All the listed wave equations with second-order derivatives in time need two initial conditions:

We introduce a mesh in time and in space. The mesh in time consists of time points normally, for wave equation problems, with a constant spacing \(\Delta t=t_{n+1}-t_{n}\), \(n\in\mathcal{I}_{t}^{-}\).

Finite difference methods are easy to implement on simple rectangle- or box-shaped spatial domains. More complicated shapes of the spatial domain require substantially more advanced techniques and implementational efforts (and a finite element method is usually a more convenient approach). On a rectangle- or box-shaped domain, mesh points are introduced separately in the various space directions: We can write a general mesh point as \((x_{i},y_{j},z_{k},t_{n})\), with \(i\in\mathcal{I}_{x}\), \(j\in\mathcal{I}_{y}\), \(k\in\mathcal{I}_{z}\), and \(n\in\mathcal{I}_{t}\).

It is a very common choice to use constant mesh spacings: \(\Delta x=x_{i+1}-x_{i}\), \(i\in\mathcal{I}_{x}^{-}\), \(\Delta y=y_{j+1}-y_{j}\), \(j\in\mathcal{I}_{y}^{-}\), and \(\Delta z=z_{k+1}-z_{k}\), \(k\in\mathcal{I}_{z}^{-}\). With equal mesh spacings one often introduces \(h=\Delta x=\Delta y=\Delta z\).

The unknown u at mesh point \((x_{i},y_{j},z_{k},t_{n})\) is denoted by \(u^{n}_{i,j,k}\). In 2D problems we just skip the z coordinate (by assuming no variation in that direction: \(\partial/\partial z=0\)) and write \(u^{n}_{i,j}\).

Two- and three-dimensional wave equations are easily discretized by assembling building blocks for discretization of 1D wave equations, because the multi-dimensional versions just contain terms of the same type as those in 1D.

Assuming, as usual, that all values at time levels n and n − 1 are known, we can solve for the only unknown \(u^{n+1}_{i,j}\). The result can be compactly written as

As in the 1D case, we need to develop a special formula for \(u^{1}_{i,j}\) where we combine the general scheme for \(u^{n+1}_{i,j}\), when n = 0, with the discretization of the initial condition:

The result becomes, in compact form,

The PDE (2.108) with variable coefficients is discretized term by term using the corresponding elements from the 1D case: When written out and solved for the unknown \(u^{n+1}_{i,j,k}\), one gets the scheme

Also here we need to develop a special formula for \(u^{1}_{i,j,k}\) by combining the scheme for n = 0 with the discrete initial condition, which is just a matter of inserting \(u^{-1}_{i,j,k}=u^{1}_{i,j,k}-2\Delta tV_{i,j,k}\) in the scheme and solving for \(u^{1}_{i,j,k}\).

Consider the condition \(\partial u/\partial n=0\) at a boundary y = 0 of a rectangular domain \([0,L_{x}]\times[0,L_{y}]\) in 2D. The normal direction is then in −y direction, so and we set From this it follows that \(u^{n}_{i,-1}=u^{n}_{i,1}\). The discretized PDE at the boundary point \((i,0)\) reads We can then just insert \(u^{n}_{i,1}\) for \(u^{n}_{i,-1}\) in this equation and solve for the boundary value \(u^{n+1}_{i,0}\), just as was done in 1D.

From these calculations, we see a pattern: the general scheme applies at the boundary j = 0 too if we just replace j − 1 by j + 1. Such a pattern is particularly useful for implementations. The details follow from the explained 1D case in Sect. 2.6.3.

The alternative approach to eliminating fictitious values outside the mesh is to have \(u^{n}_{i,-1}\) available as a ghost value. The mesh is extended with one extra line (2D) or plane (3D) of ghost cells at a Neumann boundary. In the present example it means that we need a line with ghost cells below the y axis. The ghost values must be updated according to \(u^{n+1}_{i,-1}=u^{n+1}_{i,1}\).

The solver function for a 2D case with constant wave velocity and boundary condition u = 0 is analogous to the 1D case with similar parameter values (see wave1D_u0.py), apart from a few necessary extensions. The code is found in the program wave2D_u0.py .

The key finite difference formula (2.110) for updating the solution at a time level is implemented in a separate function as

The step1 variable has been introduced to allow the formula to be reused for the first step, computing \(u^{1}_{i,j}\):

The scalar code above turns out to be extremely slow for large 2D meshes, and probably useless in 3D beyond debugging of small test cases. Vectorization is therefore a must for multi-dimensional finite difference computations in Python. For example, with a mesh consisting of 30 × 30 cells, vectorization brings down the CPU time by a factor of 70 (!). Equally important, vectorized code can also easily be parallelized to take (usually) optimal advantage of parallel computer platforms.

In the vectorized case, we must be able to evaluate user-given functions like \(I(x,y)\) and \(f(x,y,t)\) for the entire mesh in one operation (without loops). These user-given functions are provided as Python functions I(x,y) and f(x,y,t), respectively. Having the one-dimensional coordinate arrays x and y is not sufficient when calling I and f in a vectorized way. We must extend x and y to their vectorized versions xv and yv:

This is a standard required technique when evaluating functions over a 2D mesh, say sin(xv)*cos(xv), which then gives a result with shape (Nx+1,Ny+1). Calling I(xv, yv) and f(xv, yv, t[n]) will now return I and f values for the entire set of mesh points.

With the xv and yv arrays for vectorized computing, setting the initial condition is just a matter of

One could also have written u_n = I(xv, yv) and let u_n point to a new object, but vectorized operations often make use of direct insertion in the original array through u_n[:,:], because sometimes not all of the array is to be filled by such a function evaluation. This is the case with the computational scheme for \(u^{n+1}_{i,j}\):

Array slices in 2D are more complicated to understand than those in 1D, but the logic from 1D applies to each dimension separately. For example, when doing \(u^{n}_{i,j}-u^{n}_{i-1,j}\) for \(i\in\mathcal{I}_{x}^{+}\), we just keep j constant and make a slice in the first index: u_n[1:,j] - u_n[:-1,j], exactly as in 1D. The 1: slice specifies all the indices \(i=1,2,\ldots,N_{x}\) (up to the last valid index), while :-1 specifies the relevant indices for the second term: \(0,1,\ldots,N_{x}-1\) (up to, but not including the last index).

The boundary conditions along the four sides make use of a slice consisting of all indices along a boundary:

In the vectorized update of u (above), the function f is first computed as an array over all mesh points:

We could, alternatively, have used the call f(xv, yv, t[n])[1:-1,1:-1] in the last term of the update statement, but other implementations in compiled languages benefit from having f available in an array rather than calling our Python function f(x,y,t) for every point.

Also in the advance_vectorized function we have introduced a boolean step1 to reuse the formula for the first time step in the same way as we did with advance_scalar. We refer to the solver function in wave2D_u0.py for the details on how the overall algorithm is implemented.

The callback function now has the arguments u, x, xv, y, yv, t, n. The inclusion of xv and yv makes it easy to, e.g., compute an exact 2D solution in the callback function and compute errors, through an expression like u - u_exact(xv, yv, t[n]).

Eventually, we are ready for a real application with our code! Look at the wave2D_u0.py and the gaussian function. It starts with a Gaussian function to see how it propagates in a square with u = 0 on the boundaries:

Video files can be made of the PNG frames:

It is wise to use a high frame rate – a low one will just skip many frames. There may also be considerable quality differences between the different formats.

To obtain Mayavi on Ubuntu platforms you can write

For Mac OS X and Windows, we recommend using Anaconda. To obtain Mayavi for Anaconda you can write

and have plt (as usual) or mlab as a kind of MATLAB visualization access inside our program (just more powerful and with higher visual quality).

The official documentation of the mlab module is provided in two places, one for the basic functionality ¹² and one for further functionality ¹³. Basic figure handling ¹⁴ is very similar to the one we know from Matplotlib. Just as for Matplotlib, all plotting commands you do in mlab will go into the same figure, until you manually change to a new figure.

Back to our application, the following code for the user action function with plotting in Mayavi is relevant to add.

Figure 2.10 shows a model we may use to derive the equation for waves on a string. The string is modeled as a set of discrete point masses (at mesh points) with elastic strings in between. The string has a large constant tension T. We let the mass at mesh point x _i be m _i . The displacement of this mass point in the y direction is denoted by \(u_{i}(t)\).

The motion of mass m _i is governed by Newton’s second law of motion. The position of the mass at time t is \(x_{i}\boldsymbol{i}+u_{i}(t)\boldsymbol{j}\), where i and j are unit vectors in the x and y direction, respectively. The acceleration is then \(u_{i}^{\prime\prime}(t)\boldsymbol{j}\). Two forces are acting on the mass as indicated in Fig. 2.10. The force \(\boldsymbol{T}^{-}\) acting toward the point \(x_{i-1}\) can be decomposed as where ϕ is the angle between the force and the line \(x=x_{i}\). Let \(\Delta u_{i}=u_{i}-u_{i-1}\) and let \(\Delta s_{i}=\sqrt{\Delta u_{i}^{2}+(x_{i}-x_{i-1})^{2}}\) be the distance from mass \(m_{i-1}\) to mass m _i . It is seen that \(\cos\phi=\Delta u_{i}/\Delta s_{i}\) and \(\sin\phi=(x_{i}-x_{i-1})/\Delta s\) or \(\Delta x/\Delta s_{i}\) if we introduce a constant mesh spacing \(\Delta x=x_{i}-x_{i-1}\). The force can then be written The force \(\boldsymbol{T}^{+}\) acting toward \(x_{i+1}\) can be calculated in a similar way: Newton’s second law becomes which gives the component equations

A basic reasonable assumption for a string is small displacements u _i and small displacement gradients \(\Delta u_{i}/\Delta x\). For small \(g=\Delta u_{i}/\Delta x\) we have that Equation (2.120) is then simply the identity T = T, while (2.121) can be written as which upon division by \(\Delta x\) and introducing the density \(\varrho_{i}=m_{i}/\Delta x\) becomes We can now choose to approximate \(u_{i}^{\prime\prime}\) by a finite difference in time and get the discretized wave equation, On the other hand, we may go to the continuum limit \(\Delta x\rightarrow 0\) and replace \(u_{i}(t)\) by \(u(x,t)\), \(\varrho_{i}\) by \(\varrho(x)\), and recognize that the right-hand side of (2.122) approaches \(\partial^{2}u/\partial x^{2}\) as \(\Delta x\rightarrow 0\). We end up with the continuous model for waves on a string: Note that the density \(\varrho\) may change along the string, while the tension T is a constant. With variable wave velocity \(c(x)=\sqrt{T/\varrho(x)}\) we can write the wave equation in the more standard form Because of the way \(\varrho\) enters the equations, the variable wave velocity does not appear inside the derivatives as in many other versions of the wave equation. However, most strings of interest have constant \(\varrho\).

The end points of a string are fixed so that the displacement u is zero. The boundary conditions are therefore u = 0.

Consider an elastic rod subject to a hammer impact at the end. This experiment will give rise to an elastic deformation pulse that travels through the rod. A mathematical model for longitudinal waves along an elastic rod starts with the general equation for deformations and stresses in an elastic medium, where \(\varrho\) is the density, u the displacement field, σ the stress tensor, and f body forces. The latter has normally no impact on elastic waves.

For stationary deformation of an elastic rod, aligned with the x axis, one has that \(\sigma_{xx}=Eu_{x}\), with all other stress components being zero. The parameter E is known as Young’s modulus. Moreover, we set \(\boldsymbol{u}=u(x,t)\boldsymbol{i}\) and neglect the radial contraction and expansion (where Poisson’s ratio is the important parameter). Assuming that this simple stress and deformation field is a good approximation, (2.127) simplifies to

The associated boundary conditions are u or \(\sigma_{xx}=Eu_{x}\) known, typically u = 0 for a fixed end and \(\sigma_{xx}=0\) for a free end.

Think of a thin, elastic membrane with shape as a circle or rectangle. This membrane can be brought into oscillatory motion and will develop elastic waves. We can model this phenomenon somewhat similar to waves in a rod: waves in a membrane are simply the two-dimensional counterpart. We assume that the material is deformed in the z direction only and write the elastic displacement field on the form \(\boldsymbol{u}(x,y,t)=w(x,y,t)\boldsymbol{i}\). The z coordinate is omitted since the membrane is thin and all properties are taken as constant throughout the thickness. Inserting this displacement field in Newton’s 2nd law of motion (2.127) results in This is nothing but a wave equation in \(w(x,y,t)\), which needs the usual initial conditions on w and w _t as well as a boundary condition w = 0. When computing the stress in the membrane, one needs to split σ into a constant high-stress component due to the fact that all membranes are normally pre-stressed, plus a component proportional to the displacement and governed by the wave motion.

Seismic waves are used to infer properties of subsurface geological structures. The physical model is a heterogeneous elastic medium where sound is propagated by small elastic vibrations. The general mathematical model for deformations in an elastic medium is based on Newton’s second law, and a constitutive law relating σ to u, often Hooke’s generalized law, Here, u is the displacement field, σ is the stress tensor, I is the identity tensor, \(\varrho\) is the medium’s density, f are body forces (such as gravity), K is the medium’s bulk modulus and G is the shear modulus. All these quantities may vary in space, while u and σ will also show significant variation in time during wave motion.

The acoustic approximation to elastic waves arises from a basic assumption that the second term in Hooke’s law, representing the deformations that give rise to shear stresses, can be neglected. This assumption can be interpreted as approximating the geological medium by a fluid. Neglecting also the body forces f, (2.130) becomes Introducing p as a pressure via and dividing (2.132) by \(\varrho\), we get Taking the divergence of this equation, using \(\nabla\cdot\boldsymbol{u}=-p/K\) from (2.133), gives the acoustic approximation to elastic waves: This is a standard, linear wave equation with variable coefficients. It is common to add a source term \(s(x,y,z,t)\) to model the generation of sound waves:

A common additional approximation of (2.136) is based on using the chain rule on the right-hand side, under the assumption that the relative spatial gradient \(\nabla\varrho^{-1}=-\varrho^{-2}\nabla\varrho\) is small. This approximation results in the simplified equation

The acoustic approximations to seismic waves are used for sound waves in the ground, and the Earth’s surface is then a boundary where p equals the atmospheric pressure p ₀ such that the boundary condition becomes \(p=p_{0}\).

Sound waves arise from pressure and density variations in fluids. The starting point of modeling sound waves is the basic equations for a compressible fluid where we omit viscous (frictional) forces, body forces (gravity, for instance), and temperature effects: These equations are often referred to as the Euler equations for the motion of a fluid. The parameters involved are the density \(\varrho\), the velocity u, and the pressure p. Equation (2.139) reflects mass balance, (2.140) is Newton’s second law for a fluid, with frictional and body forces omitted, and (2.141) is a constitutive law relating density to pressure by thermodynamic considerations. A typical model for (2.141) is the so-called isentropic relation ¹⁵, valid for adiabatic processes where there is no heat transfer: Here, p ₀ and \(\varrho_{0}\) are reference values for p and \(\varrho\) when the fluid is at rest, and γ is the ratio of specific heat at constant pressure and constant volume (γ = 5 ∕ 3 for air).

The key approximation in a mathematical model for sound waves is to assume that these waves are small perturbations to the density, pressure, and velocity. We therefore write where we have decomposed the fields in a constant equilibrium value, corresponding to u = 0, and a small perturbation marked with a hat symbol. By inserting these decompositions in (2.139) and (2.140), neglecting all product terms of small perturbations and/or their derivatives, and dropping the hat symbols, one gets the following linearized PDE system for the small perturbations in density, pressure, and velocity: Now we can eliminate \(\varrho_{t}\) by differentiating the relation \(\varrho(p)\), The product term \(p^{1/\gamma-1}p_{t}\) can be linearized as \(p_{0}^{1/\gamma-1}p_{t}\), resulting in We then get Taking the divergence of (2.146) and differentiating (2.145) with respect to time gives the possibility to easily eliminate \(\nabla\cdot\boldsymbol{u}_{t}\) and arrive at a standard, linear wave equation for p: where \(c=\sqrt{\gamma p_{0}/\varrho_{0}}\) is the speed of sound in the fluid.

Spherically symmetric three-dimensional waves propagate in the radial direction r only so that \(u=u(r,t)\). The fully three-dimensional wave equation then reduces to the spherically symmetric wave equation One can easily show that the function \(v(r,t)=ru(r,t)\) fulfills a standard wave equation in Cartesian coordinates if c is constant. To this end, insert u = v ∕ r in to obtain The two terms in the parenthesis can be combined to which is recognized as the variable-coefficient Laplace operator in one Cartesian coordinate. The spherically symmetric wave equation in terms of \(v(r,t)\) now becomes In the case of constant wave velocity c, this equation reduces to the wave equation in a single Cartesian coordinate called r: That is, any program for solving the one-dimensional wave equation in a Cartesian coordinate system can be used to solve (2.150), provided the source term is multiplied by the coordinate, and that we divide the Cartesian mesh solution by r to get the spherically symmetric solution. Moreover, if r = 0 is included in the domain, spherical symmetry demands that \(\partial u/\partial r=0\) at r = 0, which means that For this to hold in the limit \(r\rightarrow 0\), we must have \(v(0,t)=0\) at least as a necessary condition. In most practical applications, we exclude r = 0 from the domain and assume that some boundary condition is assigned at r = ϵ, for some ϵ > 0.

The next example considers water waves whose wavelengths are much larger than the depth and whose wave amplitudes are small. This class of waves may be generated by catastrophic geophysical events, such as earthquakes at the sea bottom, landslides moving into water, or underwater slides (or a combination, as earthquakes frequently release avalanches of masses). For example, a subsea earthquake will normally have an extension of many kilometers but lift the water only a few meters. The wave length will have a size dictated by the earthquake area, which is much lager than the water depth, and compared to this wave length, an amplitude of a few meters is very small. The water is essentially a thin film, and mathematically we can average the problem in the vertical direction and approximate the 3D wave phenomenon by 2D PDEs. Instead of a moving water domain in three space dimensions, we get a horizontal 2D domain with an unknown function for the surface elevation and the water depth as a variable coefficient in the PDEs.

Let \(\eta(x,y,t)\) be the elevation of the water surface, \(H(x,y)\) the water depth corresponding to a flat surface (η = 0), \(u(x,y,t)\) and \(v(x,y,t)\) the depth-averaged horizontal velocities of the water. Mass and momentum balance of the water volume give rise to the PDEs involving these quantities: where g is the acceleration of gravity. Equation (2.151) corresponds to mass balance while the other two are derived from momentum balance (Newton’s second law).

The initial conditions associated with (2.151)–(2.153) are η, u, and v prescribed at t = 0. A common condition is to have some water elevation \(\eta=I(x,y)\) and assume that the surface is at rest: u = v = 0. A subsea earthquake usually means a sufficiently rapid motion of the bottom and the water volume to say that the bottom deformation is mirrored at the water surface as an initial lift \(I(x,y)\) and that u = v = 0.

Boundary conditions may be η prescribed for incoming, known waves, or zero normal velocity at reflecting boundaries (steep mountains, for instance): \(un_{x}+vn_{y}=0\), where \((n_{x},n_{y})\) is the outward unit normal to the boundary. More sophisticated boundary conditions are needed when waves run up at the shore, and at open boundaries where we want the waves to leave the computational domain undisturbed.

Equations (2.151), (2.152), and (2.153) can be transformed to a standard, linear wave equation. First, multiply (2.152) and (2.153) by H, differentiate (2.152) with respect to x and (2.153) with respect to y. Second, differentiate (2.151) with respect to t and use that \((Hu)_{xt}=(Hu_{t})_{x}\) and \((Hv)_{yt}=(Hv_{t})_{y}\) when H is independent of t. Third, eliminate \((Hu_{t})_{x}\) and \((Hv_{t})_{y}\) with the aid of the other two differentiated equations. These manipulations result in a standard, linear wave equation for η:

In the case we have an initial non-flat water surface at rest, the initial conditions become \(\eta=I(x,y)\) and \(\eta_{t}=0\). The latter follows from (2.151) if u = v = 0, or simply from the fact that the vertical velocity of the surface is η _t , which is zero for a surface at rest.

The system (2.151)–(2.153) can be extended to handle a time-varying bottom topography, which is relevant for modeling long waves generated by underwater slides. In such cases the water depth function H is also a function of t, due to the moving slide, and one must add a time-derivative term H _t to the left-hand side of (2.151). A moving bottom is best described by introducing \(z=H_{0}\) as the still-water level, \(z=B(x,y,t)\) as the time- and space-varying bottom topography, so that \(H=H_{0}-B(x,y,t)\). In the elimination of u and v one may assume that the dependence of H on t can be neglected in the terms \((Hu)_{xt}\) and \((Hv)_{yt}\). We then end up with a source term in (2.154), because of the moving (accelerating) bottom:

The reduction of (2.155) to 1D, for long waves in a straight channel, or for approximately plane waves in the ocean, is trivial by assuming no change in y direction (\(\partial/\partial y=0\)):

The flow of blood in our bodies is basically fluid flow in a network of pipes. Unlike rigid pipes, the walls in the blood vessels are elastic and will increase their diameter when the pressure rises. The elastic forces will then push the wall back and accelerate the fluid. This interaction between the flow of blood and the deformation of the vessel wall results in waves traveling along our blood vessels.

A model for one-dimensional waves along blood vessels can be derived from averaging the fluid flow over the cross section of the blood vessels. Let x be a coordinate along the blood vessel and assume that all cross sections are circular, though with different radii \(R(x,t)\). The main quantities to compute is the cross section area \(A(x,t)\), the averaged pressure \(P(x,t)\), and the total volume flux \(Q(x,t)\). The area of this cross section is Let \(v_{x}(x,t)\) be the velocity of blood averaged over the cross section at point x. The volume flux, being the total volume of blood passing a cross section per time unit, becomes

Mass balance and Newton’s second law lead to the PDEs where γ is a parameter related to the velocity profile, \(\varrho\) is the density of blood, and μ is the dynamic viscosity of blood.

We have three unknowns A, Q, and P, and two equations (2.159) and (2.160). A third equation is needed to relate the flow to the deformations of the wall. A common form for this equation is where C is the compliance of the wall, given by the constitutive relation which requires a relationship between A and P. One common model is to view the vessel wall, locally, as a thin elastic tube subject to an internal pressure. This gives the relation where P ₀ and A ₀ are corresponding reference values when the wall is not deformed, h is the thickness of the wall, and E and ν are Young’s modulus and Poisson’s ratio of the elastic material in the wall. The derivative becomes Another (nonlinear) deformation model of the wall, which has a better fit with experiments, is where β is some parameter to be estimated. This law leads to

Light and radio waves are governed by standard wave equations arising from Maxwell’s general equations. When there are no charges and no currents, as in a vacuum, Maxwell’s equations take the form where \(\epsilon_{0}=8.854187817620\cdot 10^{-12}\) (F/m) is the permittivity of free space, also known as the electric constant, and \(\mu_{0}=1.2566370614\cdot 10^{-6}\) (H/m) is the permeability of free space, also known as the magnetic constant. Taking the curl of the two last equations and using the mathematical identity gives the wave equation governing the electric and magnetic field: with \(c=1/\sqrt{\mu_{0}\epsilon_{0}}\) as the velocity of light. Each component of E and B fulfills a wave equation and can hence be solved independently.