High order finite difference methods for the wave equation with non-conforming grid interfaces

Abstract

We use high order finite difference methods to solve the wave equation in the second order form. The spatial discretization is performed by finite difference operators satisfying a summation-by-parts property. The focus of this work is on the numerical treatment of non-conforming grid interfaces. The interface conditions are imposed weakly by the simultaneous approximation term technique in combination with interface operators, which move the discrete solutions between the grids on the interface. In particular, we consider interpolation operators and projection operators. A norm-compatibility condition, which leads to stability for first order hyperbolic systems, does not suffice for second order wave equations. An extra constraint on the interface operators must be satisfied to derive an energy estimate for stability. We carry out eigenvalue analyses to investigate the additional constraint and how it is related to stability, and find that the projection operators have better stability properties than the interpolation operators. In addition, a truncation error analysis is performed to study the convergence property of the numerical schemes. In the numerical experiments, the stability and accuracy properties of the numerical schemes are further explored, and the practical usefulness of non-conforming grid interfaces is presented and discussed in two efficiency studies.

27 pages, 15 figures

References in corpus (4)

• Review of Summation-by-parts schemes for initial-boundary-value problems
• Stable Coupling of Nonconforming, High-Order Finite Difference Methods
• Stable difference methods for block-oriented adaptive grids
• Convergence of summation-by-parts finite difference methods for the wave equation

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• A Non-stiff Summation-By-Parts Finite Difference Method for the Scalar Wave Equation in Second Order Form: Characteristic Boundary Conditions and Nonlinear Interfaces
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• Summation-by-parts operators for general function spaces: The second derivative
• Mortar-based entropy-stable discontinuous Galerkin methods on non-conforming quadrilateral and hexahedral meshes
• Fourth order finite difference methods for the wave equation with mesh refinement interfaces
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• Simultaneous approximation terms for elastic wave equations on nonuniform grids
• Explicit coupling of acoustic and elastic wave propagation in finite difference simulations