Numerical method satisfying the first two conservation laws for the Korteweg–de Vries equation

Abstract

In this paper, we develop a finite-volume scheme for the KdV equation which conserves both the momentum and energy. The main ingredient of the method is a numerical device we developed in recent years that enables us to construct numerical method for a PDE that also simulates its related equations. In the method, numerical approximations to both the momentum and energy are conservatively computed. The operator splitting approach is adopted in constructing the method in which the conservation and dispersion parts of the equation are alternatively solved; our numerical device is applied in solving the conservation part of the equation. The feasibility and stability of the method is discussed, which involves an important property of the method, the so-called Jensen condition. The truncation error of the method is analyzed, which shows that the method is second-order accurate. Finally, several numerical examples, including the Zabusky–Kruskal’s example, are presented to show the good stability property of the method for long-time numerical integration.

Introduction

It is well known that the Korteweg–de Vries (KdV) equation

$$u_t+{\left(\frac{1}{2}{u}^2\right)}_x+\epsilon u_{\mathit{xxx}}=0\text{,}$$

where ε is a constant, possesses an infinite set of conservation laws and the first two of them are Eq. (1.1), which describes the conservation of momentum and

$$(u^2{)}_t+{\left(\frac{2}{3}{u}^3\right)}_x+\epsilon (2{\mathit{uu}}_{\mathit{xx}}-(u_x{)}^2{)}_x=0\text{,}$$

which describes the conservation of energy. It is commonly believed that it is good for a numerical method for the KdV equation to simulate as many of these conservation relations as possible; methods preserving more conservation relations are usually more stable and suitable for long-time integration. However, maintaining more than one conservation relations is difficult in the practice of numerical simulation. From a broader point of view of mathematics, the difficulty is that there is generally no enough degrees of freedom to construct numerical methods for a PDE that also simulates the equations that are derived from the original one.

A lot of numerical methods have been developed for the KdV equation in recent years, ranging from finite difference methods, finite element methods to spectral methods, see [25], [8], [16], [24], [23], [3], [1], [2], [20], [26], [5], [6] and the references cited therein. Most of them preserve the conservation of momentum and only a few of them preserve the conservation of both the momentum and energy. We should particularly mention the symplectic and multisymplectic schemes, see [3], [1], [2], [26], [20]. When viewing the KdV equation as a Hamiltonian system, these schemes preserve the symplectic structure of the system. Therefore, they are proved to be good numerical methods for long-time integration of the equation. However, symplectic and multisymplectic schemes usually do not exactly conserve the momentum and energy and thus how closely the momentum and energy are conserved in computation is then a property of interest for these schemes and is often assessed in numerical experiments, see the references cited above.

In this paper, we develop a finite difference method for the KdV equation which satisfies both the momentum and energy conservation relations. The main ingredient of the method is a device we developed in recent years that enables us to construct numerical methods that simultaneously simulates a PDE and its related equations, see [11], [4], [12], [13], [14], [21], [22]. To explain the idea, we look at the linear advection equation

$$u_t+u_x=0\text{.}$$

This equation possesses infinitively many conservation laws. As a matter of fact, for any smooth function U(u) the following equation

$$U(u{)}_t+U(u{)}_x=0$$

is also satisfied. Both u and U(u) are then conserved in the sense that

$${\int }_{-\infty }^{\infty }u(x\text{,}t)\mathrm{d}x={\int }_{-\infty }^{\infty }u(x\text{,}0)\mathrm{d}x$$

and

$${\int }_{-\infty }^{\infty }U(u(x\text{,}t))\mathrm{d}x={\int }_{-\infty }^{\infty }U(u(x\text{,}0))\mathrm{d}x\text{.}$$

In the following, we are going to construct a conservative numerical scheme that simulates both Eqs. (1.3), (1.4) for a given U(u).

Our scheme is of the Godunov type and its numerical solution $\{u_j^n\}$ is a cell-average approximation to the exact solution at time t_n

$$u_j^n\simeq \frac{1}{h}{\int }_{x_{j-\frac{1}{2}}}^{x_{j+\frac{1}{2}}}u(x\text{,}{t}_n)\mathrm{d}x\text{.}$$

A very special feature of the scheme is that it computes also a cell-average approximation to U(u)

$$U_j^n\simeq \frac{1}{h}{\int }_{x_{j-\frac{1}{2}}}^{x_{j+\frac{1}{2}}}U(u(x\text{,}{t}_n))\mathrm{d}x\text{.}$$

Like all the Godunov-type schemes (see [9], [10]), the scheme proceeds in the reconstruction, evolution and cell-averaging steps.

In the reconstruction step, the solution is reconstructed in each cell $(x_{j-\frac{1}{2}}\text{,}{x}_{j+\frac{1}{2}})$ as a linear function

$$R(x\text{;}{u}^n\text{,}{U}^n)=u_j^n+s_j^n(x-x_j)\text{,}j=\dots \text{,}-1\text{,}0\text{,}1\text{,}\dots \text{,}$$

where $s_j^n$ is the slope. Another very special feature of the scheme is that the slope is not computed by interpolating the solution as is done in ordinary finite-volume schemes, but rather by requiring

$$\frac{1}{h}{\int }_{x_{j-\frac{1}{2}}}^{x_{j+\frac{1}{2}}}U(R(x\text{;}{u}^n\text{,}{U}^n))\mathrm{d}x=U_j^n\text{,}$$

that is, the cell-average of U(u) of the reconstructed solution must be equal to the numerical cell-average of U(u) in the grid cell. Eq. (1.10) is an equation of $s_j^n$, from which we solve out the slope.

The evolution step is to solve the IVP

$$\left\{\begin{array}{c}{v}_t+v_x=0\text{,}\hfill \\ v(x\text{,}{t}_n)=R(x\text{;}{u}^n\text{,}{U}^n)\text{,}\hfill \end{array}\right.t_n<t<t_{n+1}\text{,}$$

as that in all the Godunov’s type schemes do and the cell-averaging step is to cell-average v and U(v) to Eq. (1.11) at t = t_n+1 to obtain $u_j^{n+1}$ and $U_j^{n+1}$. In practice, we use the integral form of Eqs. (1.3), (1.4) to compute $u_j^{n+1}$ and $U_j^{n+1}$, which results in the following conservative schemes

$$u_j^{n+1}=u_j^n-\lambda ({\hat{f}}_{j+1/2}^n-{\hat{f}}_{j-1/2}^n)$$

and

$$U_j^{n+1}=U_j^n-\lambda ({\widehat{F}}_{j+1/2}^n-{\widehat{F}}_{j-1/2}^n)\text{,}$$

where ${\hat{f}}_{j+1/2}^n$ and ${\widehat{F}}_{j+1/2}^n$ are the flux approximations to u and U(u) in Eqs. (1.3), (1.4), respectively, on the cell boundaries.

The scheme so constructed maintains the conservation for both u and U(u). We note that our numerical approximations $u_j^n$ and $U_j^n$ are not related as customarily $U(u_j^n)=U_j^n$, but rather in a loosen fashion as in Eqs. (1.8), (1.10). Actually, the numerical solution in each grid cell can be understood from Eqs. (1.8), (1.10) as a piece of linear function whose cell-average is $u_j^n$ and whose cell-average of U(u) is $U_j^n$. In this fashion, we gain one degree of freedom in describing the numerical solution, which enables us to maintain the conservation relations for both u and U(u) in constructing the scheme.

Along this line, schemes maintaining more conservation relations can also be constructed. This can be accomplished by involving more entities $(U_1{)}^n\text{,}(U_2{)}^n\text{,}\dots$ in the numerical scheme, where

$$(U_i{)}_j^n\simeq \frac{1}{h}{\int }_{x_{j-\frac{1}{2}}}^{x_{j+\frac{1}{2}}}{U}_i(u(x\text{,}{t}_n))\mathrm{d}x\text{,}$$

with U_i(u) being nonlinear functions of u, either convex or not, reconstructing the solution in each cell as polynomial

$$R(x\text{;}{u}^n\text{,}(U_1{)}^n\text{,}(U_2{)}^n\text{,}\dots )=u_j^n+s_1(x-x_j)+s_2(x-x_j{)}^2+\cdots \text{,}$$

and solving the coefficients s₁, s₂, … from the system of equations

$$\frac{1}{h}{\int }_{x_{j-\frac{1}{2}}}^{x_{j+\frac{1}{2}}}{U}_i(R(x\text{;}{u}^n\text{,}(U_1{)}^n\text{,}(U_2{)}^n\text{,}\dots ))\mathrm{d}x=(U_i{)}_j^n\text{,}i=1\text{,}2\text{,}\dots \text{,}$$

see [21]. For more general evolution PDE’s, the functions U_i’s may also involve the derivatives of u, i.e. u_x,$u_{\mathit{xx}}\text{,}\dots$ and U_i’s are not necessary to be conserved.

For continuous solutions to Eq. (1.3), the numerical results computed by this kind of schemes are fantastic. In Fig. 1, we present two numerical results computed by this kind of schemes with 200 grid cells, which are cited from [21]. Both are of the Wavepacket problem, see [9]. The result on the left is obtained by a scheme maintaining u and u² conserved (second-order accurate) and is at t = 200 and the result on the right is by a scheme maintaining u, u² and u³ conserved (third-order accurate) and is at t = 20,000. As to our knowledge, no scheme up to date has ever got qualified numerical result for this example with the same grid beyond the time t = 20.

The success of our numerical device in the linear advection equation encourages us to apply it to the KdV equation since the latter also possesses many conservation laws and maintaining them in numerical simulation is important. We develop a scheme which computes and conserves both the momentum u and energy U(u). That is, the constructed scheme involves two numerical entities, the numerical momentum $u_j^n$ and numerical energy $U_j^n$ and both of them are conservatively computed.

To construct the scheme, we adopt the splitting strategy as in [8], i.e. split the Eq. (1.1) into the conservation part (2.5) and dispersion part (2.6). Our numerical device is applied in solving the conservation part, where the solution is reconstructed in each grid cell with its momentum cell-average being $u_j^n$ and its energy cell-average being $U_j^n$. To solve the dispersion part, we adopt the implicit difference scheme in [8] to compute the momentum cell-averages $\{u_j^n\}$ and once they are computed the energy cell-averages $\{U_j^n\}$ are then computed passively from them. The advantage of using this implicit scheme is that the time step τ does not suffer from a prohibiting restriction τ = O(h³), and the corresponding linear algebra system is still easy to be solved using well developed numerical methods. Therefore, the only restriction on the time step in our method comes from the conservation part, which is of τ = O(h).

Constructed in such a way, our method is stable in the sense that its energy is nonnegative and L¹ bounded and its momentum is L² bounded by the L¹ norm of its energy, see Theorem 3.2. The method is second-order accurate away from extremes of solution and is at least first-order accurate near extremes, see Theorem 4.3.

We know that the KdV equation is notorious for its “marginal” stability that is resulted in the balance between the nonlinear convection and the linear dispersion, see [2]. This “living at the edge of stability” becomes more fragile in the “convection-dominated” cases, in which the linear dispersion is very weak compared to the nonlinear convection; therefore, long-time numerical integrations for these cases are difficult since they tend to become unstable. A typical case is the so-called Zabusky–Kruskal’s example, see [25], on which many numerical schemes, including the Zabusky–Kruskal’s scheme, explode in long-time numerical integrations, see [1], [18], [20], [26], [5]. We test our method on “convection-dominated” cases, especilally the Zabusky–Kruskal’s example and the numerical results are fantastic. The method is very stable and robust in long-time integrations and the solitions are very smooth with well-preserved shapes of solitons. The numerical simulation for the Zabusky–Kruskal’s example goes up to the time t = 20t_R and still does not exhibit any blowup. It seems to us that the computation can go on for ever.

The organization of the paper is as follows: Section 1 is the introduction. In Section 2, we describe the numerical method in detail. Section 3 discusses the feasibility and stability of the method, which involves a very important property of the method, the so-called Jensen condition. Section 4 discusses the accuracy of the method and proves that the method is essentially second-order accurate. In Section 5, we present several numerical examples computed with our method, which include the Zabusky–Kruskal’s example [25], to show its good stability property for long-time numerical integration. Finally, Section 6 is the conclusion.