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Weber, Ivy; Kreiss, Gunilla; Nazarov, Murtazo

Stability analysis of high order methods for the wave equation. (English) Zbl 1524.65611

J. Comput. Appl. Math. 404, Article ID 113900, 20 p. (2022).
Summary: In this paper, we investigate the stability of a numerical method for solving the wave equation. The method uses explicit leap-frog in time and high order continuous and discontinuous (DG) finite elements using the standard Lagrange and Hermite basis functions in space. Matrix eigenvalue analysis is used to calculate time-step restrictions. We show that the time-step restriction for continuous Lagrange elements is independent of the nodal distribution, such as equidistributed Lagrange nodes and Gauss-Lobatto nodes. We show that the time-step restriction for the symmetric interior penalty DG schemes with the usual penalty terms is tighter than for continuous Lagrange finite elements. Finally, we conclude that the best time-step restriction is obtained for continuous Hermite finite elements up to polynomial degrees \(p = 13\).

Cited in 1 Document

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
35L10 Second-order hyperbolic equations

Keywords:

time-step stability; finite elements; discontinuous Galerkin; wave equation
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