Stability of Discontinuous Galerkin Spectral Element Schemes for Wave Propagation when the Coefficient Matrices have Jumps

To motivate (and simplify) the general formulation, we start with the scalar advection equation with two domains as an introduction. Our discussion in this section restates that of [13], but introduces our notation used in succeeding sections.

We derive the energy dynamics of the solution to the scalar advection initial-boundary-value problem in the form (1)where\(a_{L}, a_{R}\) are constants, and \(a_{L}\ne a_{R}\). The discussion that follows leads to the same types of conclusions if the wave speeds are both negative. We are interested here in problems where the domains couple and waves propagate from one side to the other. So we do not consider \(a_{L}>0,a_{R}<0\), where the domains decouple as energy is dissipated at the interface, or \(a_{L}<0,a_{R}>0\) where boundary conditions for both sides are required.

We split the problem into two: Left,and Rightwhere \(u_{*}\) is the upwind specified interface condition chosen so that the Rankine–Hugoniot (or conservation) conditionis satisfied. Thus, for the scalar equation, \(u_{*}(t) =\frac{a_{L}}{a_{R}} u(0^{-},t)\).

To find the energy equation, we multiply by the solution and integrate over the domains. Define the \(L_{2}\) energy normsThenAdding together and re-arranging,where \(\left| \left| \cdot \right| \right| ^{2} = \left| \left| \cdot \right| \right| _{L}^{2} + \left| \left| \cdot \right| \right| _{R}^{2}\). Applying the homogeneous boundary condition on the left,When we apply the interface condition,The quantity Q will be used later in this paper to define stability.

Finally, we substitute the interface value for \(u_{*}\),and rearrange so thatEquation (21) shows that the energy is dissipated by the interface only if \(a_{R} > a_{L}\). Otherwise the interface generates energy, as illustrated in Fig. 3.

In [13] it was shown that one can construct “discounted norms”, in which the energy is bounded. If the second equation in (16) is multiplied by a constant \(\alpha _{c}>0\), then the weighted sum leads toThen defining the new norm with the \(\alpha _{c}\) discount factor, we haveprovided that

The discounted norm is equivalent to the \(L_2\) norm. In the discounted situation, where \(\alpha _c < 1\),andTherefore,Equivalence of the norms means that the \(L_2\) norm is actually bounded in terms of the initial data, even though the energy method does not show it directly through (21). From (23) and (31),Thus,for the constant coefficient problem.

We now increase the complexity and extend the scalar one-dimensional analysis to the general system (1) in one space dimension. We derive the energy equation for the one-dimensional hyperbolic systemwhere the coefficient matrices are for now assumed to be symmetric. Under this assumption, there is a matrix \( {P}\) such that \( {A} = {P} \varLambda {P}^{-1}\) satisfying \( {P}^{-1}= {P}^{T}\). For the moment, let us assume that \( {A}\) has no zero eigenvalues. We also assume that the number of positive and negative eigenvalues does not change across the interface. In other words, there is no eigenvalue that changes sign at the jump. Depending on the sign change, boundary/interface conditions are either lost or gained. More general conditions where the sign of the eigenvalues changes in multi-physics applications are considered in [5]. Finally, we assume that appropriate boundary and initial data are applied.

To find the interface condition at \(x=0\) for the system (34), we split the system into right and left going waves. The characteristic variables for the system (34) arewhere is associated with the positive eigenvalues of \( {A}\) and \(\mathbf{w}^{-}\) is associated with the negative ones. They are chosen upwind at the interface according towhere here and in the following, the subscripts R and L correspond to the values at \(x=0^{+}\) and \(x=0^{-}\), respectively.

The \(\mathbf{w}^{\pm }_{*}\) are computed so that the Rankine–Hugoniot conditionis satisfied at the stationary interface. Let us writeThen (37) can be written asLet us put the unknowns on the left, and the knowns on the right, givingEquation (40) provides a system of equations for the unknowns.

The matrices on the left of (40) have a special structure since \( {P}\) is the matrix of right eigenvectors and \({\varLambda }\) is a diagonal matrix. Let n be the number of positive eigenvalues out of a total of m. Then defineand where is the eigenvector associated with the eigenvalue \(\lambda _{j}\) and the eigenvalues are ordered in decreasing order, largest to smallest with \(\lambda _{j}>0\) for \(j\le n\). Then we can write (40) asGiven the structure of the \( {M}^{\pm }\) matrices, the equations can be combined to produce a single system for the unknownswhereExistence and uniqueness of the inflow characteristic vectors \(\mathbf{w}_{*}^{\pm }\) therefore depends on the existence of the inverse of the matrix \( {M}_{LR}\). That matrix is comprised of eigenvectors of the coefficient matrix evaluated on the left and eigenvectors evaluated on the right. On the one hand, if the eigenvectors of the coefficient matrix do not change across the material discontinuity, then, since the eigenvectors are independent, \( {M}_{LR}^{-1} {M}_{LR}\) is diagonal. As an example, the eigenvectors of the acoustic wave system (3) are constant, being independent of the material properties on either side. On the other hand, if the eigenvectors change across the interface and the matrix \( {M}_{LR}^{-1} {M}_{RL}\) is not diagonal, then the problem is ill-posed [5]. We therefore require that the eigenvectors be preserved across the jumps so that \( {M}_{LR}^{-1}\) exists, \( {M} \equiv {M}_{LR}^{-1} {M}_{LR}\) is diagonal, and

Going back to the original equations, (34), we compute the energy equation by multiplying by the state and integrating over the domain, givingwhere, now, \(\left| \left| \mathbf{u}\right| \right| ^{2} = \left\langle \mathbf{u},\mathbf{u}\right\rangle \) and \({\,{\text {PBT}}}\) represents the terms coming from the physical boundary conditions on the left and right. Since we are only interested here in the interface conditions, we will assume that the physical boundary conditions are well posed so that \({\,{\text {PBT}}}\ge 0\). In that case,whereis the interface contribution to the energy.

Following the steps in the scalar analysis, we now apply the interface boundary conditions on Q. We decompose the system into characteristic variables. Then we use the fact that \( {A}\) is symmetric, making \( {P}^{-1}= {P}^{T}\). With this decomposition,taking into account the upwinding of the characteristic variables, (36).

We now gather the right-going and left-going wave contributions (c.f. (19)),and then use the fact that \(\bar{{\varLambda }}^{-}<0\), to get the final form of the interface contribution, which we write in terms of its characteristic components,Equation (52) is the system version of the scalar interface condition seen in (19).

As in the scalar problem, one can construct a discounted norm \(\left| \left| \cdot \right| \right| _{B}\) for which the associated interface term \(Q_{B}\) is non-positive and the discounted norm is bounded when the coupling matrix, \( {M}\), exists and is diagonal. For instance, one simple choice is to letwhere the entries with \(\mu ^{\pm }>0\) are counted according to the number of positive and negative eigenvalues of \( {A}\). Then multiplying the system on \(x>0\) by \( {B}\) from the left, defining the norm , and following the same steps leading to (52), the interface contribution to the energy isOne then only needs to find \(\mu ^{+}\) small enough and \(\mu ^{-}\) large enough to ensure that \(Q_{B}\le 0\), in which case the new energy is bounded in terms of the initial data. Since the coupling matrix \( {M}\) is diagonal, let us split it asso that and . Then \(Q_{B}\le 0\) if \(\mu ^{\pm }\) are chosen so that

The results of the previous two sections extend to general geometries and non-symmetric coefficient matrices. In preparation for the generalization of (52), we note that within each subdomain the coefficient matrices are constant, and therefore we can re-write (1) in a split form asWe also define the inner product and norm over a subdomain \(D = {\varOmega }_{L}\) or \( {\varOmega }_{R}\) asso thatTo form the energy, we take the inner product of (57) with the vector , givingLet us define to be the symmetric system state. Then since \( {S}\) is constant within the subdomains and \( {A}^{s}= {S}^{-1} {A} {S} \),We then apply multidimensional integration by parts and symmetry to the divergence termwhere is the outward normal at the boundary of D, and note that the volume term cancels the third term in (61), leaving only the boundary integral,Then over the domain \({\varOmega }\),where L/R represent the states on either side of the interface with respect to the normal .

We can now get a bound for the multidimensional system similar to (48). The integrand in the interface integral is identical to that in (47), with and \(\mathbf{u} \leftarrow \mathbf{u}^{s}\). Therefore, if the boundary conditions along \({\varGamma }_{b}\) are properly posed and dissipative,Therefore, Q is still given by (52), but now formulated in the new symmetrized variables. Note that the norm defined by \(\left| \left| \mathbf{u}^{s}\right| \right| ^{2} = \left\langle \left( {S}^{-1}\right) ^{T} {S}^{-1}\mathbf{u},\mathbf{u}\right\rangle \) is equivalent to the norm \(\left| \left| \mathbf{u}\right| \right| \) since \(\left( {S}^{-1}\right) ^{T} {S}^{-1}>0\).

In summary, for hyperbolic systems of the form (1), with discontinuities in the coefficient matrices and homogeneous, dissipative boundary conditions, the \(L_{2}\) norm of the solution obeys (65). The integrand of the interface contribution, Q, is of the form (52), where the characteristic variables are evaluated from the upwind side and satisfy the Rankine–Hugoniot condition. It is not necessarily non-negative, depending on the relative wave speeds from either side of the interface, so the \(L_{2}\) norm of the solution is not bounded in general by the initial data. An example of such behavior was shown in Fig. 3.

Although the \(L_{2}\) norm (or, for that matter, weighted norms, see Remark 2.4) is not always bounded in terms of the initial data, there exists an energy in a discounted norm that is bounded in the usual way provided that the coupling matrix between the upwind and downwind states is diagonal.

Thus, we have two views of stability at our disposal, which we will call direct and inferred:In general geometries it may not be easy to find the discounted norm in which the solution is bounded. Finding the precise coefficients requires satisfying conditions like (56). When multiple subdomains exist in multiple space dimensions, T-type intersections between materials are possible. The discount factors must then take into account all subdomain boundaries and be adjusted globally so that at each interface (56) is still satisfied. For these reasons, it is easier to monitor the behavior (65) of the simpler \(L_{2}\) norm as a surrogate to infer well-posedness of the system. Further insights into how choosing the norm affects how the energy is bounded or not can be found in [14].

Stability of a numerical approximation of a system follows that of the PDE, and so we state the stability condition for the approximation as:

For a detailed derivation of the discrete stability estimate, which parallels the continuous analysis, we refer to [4, 21]. Here, we will only sketch some important intermediate steps. To get the stability estimate, we replace the test function \(\varvec{\varphi }\) with the approximate solution polynomial and the symmetrizer matrices, writing \(\varvec{\varphi }= \left( {S}^{-1}\right) ^{T} {S}^{-1}\mathbf{U} = \left( {S}^{-1}\right) ^{T}\mathbf{U}^s \) to getwhere we define the symmetrized discrete flux \(\mathbf{F}^s_n\) that uses the symmetric coefficient matrices . Using the fact that the symmetrizer matrix \( {S}\) commutes with the metric block matrix \(\mathfrak M\) (see e.g. [4, 21]) we see that the volume terms cancel, leaving only surface terms,When we sum over all elements, inner surface terms appear twice (with different normal vectors), whereas element surfaces that are at the physical domain boundary appear only once and are denoted as physical boundary terms (\({\,{\text {PBT}}}\)). The interior element surface contributions split into two parts: Surfaces that fall on the material interface \({\varGamma }\), and those across which the coefficient matrices are the same, which we call smooth interface boundary terms, \({\text {SIBT}}\). The sum over all elements can then be written aswritten in terms of the jump operator, . Assuming that the discrete physical boundary terms are dissipative, the discrete \(L_{2}\) norm satisfieswhich mimics the continuous stability (65) if \({\text {SIBT}}\le 0\).

We thus need a proper numerical flux function \(\mathbf{F}_n^{s,*}\) to control discrete stability, i.e. to guarantee that the integrand satisfiespointwise at each node on element faces along the discretization of \({\varGamma }\), and \({\text {SIBT}}\le 0\).

The dissipativity of the \({\text {SIBT}}\) for the upwind numerical flux has been shown elsewhere, e.g. [8, 21]. Therefore, in the following we will assume \({\text {SIBT}}\le 0\) and concern ourselves only with the discontinuous interface terms.

In the DG approximation, the Rankine–Hugoniot condition and the inflow boundary condition are enforced weakly with the upwind numerical fluxIf summation by parts is applied again to the second term in (70), one gets the strong form of the approximation, in which the integrand of the boundary term is [21]As the solution converges, this difference goes to zero, and the Rankine–Hugoniot condition is satisfied. Furthermore,so that when the approximation converges, the analytical inflow boundary condition, \(U_{R} = \frac{a_{L}}{a_{R}}U_{L}\) is approached, as required, c.f. (14).

With (76), the interface contribution for the scalar problem isFactoring the quadratic,The quadratic \(\tilde{Q}(\eta ;\frac{a_{L}}{a_{R}})\) is concave up and has a minimum when \(\eta ^{*} = a_{L}/a_{R}\), sinceWhen \(\eta ^{*} = a_{L}/a_{R}\), the Rankine–Hugoniot condition is satisfied by the states on either side. The value of that minimum is \(\tilde{Q}(\eta ^{*};\frac{a_{L}}{a_{R}}) = 1 - \frac{a_{L}}{a_{R}} \).

It then follows that the contribution to the energy in the numerical approximation matches that of the PDE, (21), plus a dissipation term dependent on how much the Rankine–Hugoniot condition is not satisfied by the approximate solution. If we define \(\beta = a_{L}/a_{R}\), and note that the minimum value of \(\tilde{Q}\) is \(1-\beta \), we can separate out that term givingRe-writing the interface contribution in the final form of (82) will be a key step in showing inferred stability of the approximation for the more complex case of a system of equations.

When we substitute for \(\eta \) and \(\beta \),Let us compare: In the continuous case, we have (21), withwhereas discretely,Thus, according to the definition of stability, Definition 2.1, the DGSEM approximation of the scalar problem with the upwind numerical flux has inferred stability.

We now parallel Sect. 2.2 and extend the analysis to a symmetric PDE system in one space dimension. For the system, the DG approximation has the interface contributionThe upwind numerical flux is nowwith the equalities between the left and right representations arising by virtue of the Rankine–Hugoniot condition. The key observation is thatBut and for the symmetric system \( {P} ^{T} {P} = {I}\), soNow,andTherefore,Looking at the second jump term in (89),soTherefore, forming \(Q_{N}\) and gathering right and left going wave components,Terms cancel, leavingFollowing (82), we now add and subtract terms to match the PDE form, which isto writeNow, letThenTo show that the approximation is stable according to Definition 2.1, then, we just need to show that \(R^{\pm }\ge 0\), since the other terms match those of the PDE. To do so, let . ThenSimilarly,Thus, the interface contribution matches that of the PDE plus an additional dissipation and has inferred stability, satisfying Definition 2.1 with

As in the continuous problem, we use the analysis of the one-dimensional problem to imply stability of the multidimensional one. As before, replace \(\mathbf{U} \leftarrow \mathbf{U}^{s}\) and . Then \(Q_{N}\) is given by (106), with the eigenvalues (and eigenvectors to construct the characteristic variables) coming from . Therefore the approximation to the general multidimensional problem is stable according to Definition 2.1.