High-Order Summation-By-Parts Finite Differences for the Second-Order Wave Equation with Piecewise Smooth Wave Speed

  • Eriksson, Gustav (Uppsala Universitet)

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It is well known that high-order finite difference methods (HOFDM) are well suited for long-time simulations of wave propagation problems. The main difficulty in constructing HOFDM lies in the treatment of boundaries and interfaces. An established approach for obtaining energy stable and robust schemes is summation-by-parts (SBP) finite differences along with the simultaneous-approximation-term (SAT) method for imposing boundary and interface conditions. Although the SBP-SAT method has been very successful in constructing energy stable discretizations of various problems in the past, for some problems the spectral radius of the discretization matrices can become very large. Resulting in inefficient time stepping schemes. In this work, the boundary and interface conditions are imposed using an alternative method based on orthogonal projections (SBP-P). With SBP-P, the stability proofs are significantly simpler than with SBP-SAT. And numerical experiments on the second-order wave equation and the dynamic beam equation have shown that the spectral radius with SBP-P is equal to or smaller than the corresponding SBP-SAT discretizations. As a model problem, the two-dimensional second-order wave equation with piecewise smooth wave speed is considered. The discontinuous wave speed is treated by splitting the computational domain into blocks and coupling the blocks together using the SBP-P method. The resulting scheme is proven energy stable in the semi-discrete setting (energy conservative with zero forcing). Furthermore, it is shown that the discretization matrix approximating the Laplacian is self-adjoint by construction. Numerical experiments are done to verify the stability properties, the order of convergence, and the spectral radius of the discretization.