Abstract
In the hyperbolic community, discontinuous Galerkin (DG) approaches are mainly applied when finite element methods are considered. As the name suggested, the DG framework allows a discontinuity at the element interfaces, which seems for many researchers a favorable property in case of hyperbolic balance laws. On the contrary, continuous Galerkin methods appear to be unsuitable for hyperbolic problems and there exists still the perception that continuous Galerkin methods are notoriously unstable. To remedy this issue, stabilization terms are usually added and various formulations can be found in the literature. However, this perception is not true and the stabilization terms are unnecessary, in general. In this paper, we deal with this problem, but present a different approach. We use the boundary conditions to stabilize the scheme following a procedure that are frequently used in the finite difference community. Here, the main idea is to impose the boundary conditions weakly and specific boundary operators are constructed such that they guarantee stability. This approach has already been used in the discontinuous Galerkin framework, but here we apply it with a continuous Galerkin scheme. No internal dissipation is needed even if unstructured grids are used. Further, we point out that we do not need exact integration, it suffices if the quadrature rule and the norm in the differential operator are the same, such that the summation-by-parts property is fulfilled meaning that a discrete Gauss Theorem is valid. This contradicts the perception in the hyperbolic community that stability issues for pure Galerkin scheme exist. In numerical simulations, we verify our theoretical analysis.
Highlights
In recent years, significant efforts have been made to construct and develop high-order methods for hyperbolic balance laws, and most of the methods are either based on finite difference (FD) or finite element (FE) approaches
What are the reasons for this belief? In our opinion, one of the major issues is that the chosen quadrature rule in the numerical integration differs from the the one used in the differential operators and without artificial stabilization terms the continuous Galerkin scheme collapses, and the corresponding Q matrix does not become almost skew-symmetric
We have demonstrated that a pure continuous Galerkin scheme is stable only through the applied boundary conditions
Summary
Significant efforts have been made to construct and develop high-order methods for hyperbolic balance laws, and most of the methods are either based on finite difference (FD) or finite element (FE) approaches. The SBP-SAT technique is powerful and universally applicable as we will show in this paper Another reason for the popularity of DG is that the numerical solution is allowed to have a discontinuity at the element boundaries, and, since non-linear hyperbolic problems are supporting shocks, this property is believed to be desirable. The difference between a DG approach and continuous Galerkin (CG), besides the structure of the mass matrix, is that in CG the approximated solution is forced to be continuous over the element boundaries This restriction is perceived to be quite strong in terms of stability where the erroneous (as we will show) belief in the hyperbolic research community exists, that a pure CG scheme is unstable, and stabilization terms have to be applied to remedy this issue [15,16,17].
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