Fourth Order Finite Difference Methods for the Wave Equation with Mesh Refinement Interfaces

Journal Article · Tue Oct 22 04:00:00 EDT 2019 · SIAM Journal on Scientific Computing

DOI:https://doi.org/10.1137/18m1211465· OSTI ID:1734602

Wang, Siyang ^[1]; Petersson, N. Anders ^[2]

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1. Chalmers Univ. of Technology, Gothenburg (Sweden); Univ. of Gothenburg (Sweden)
2. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Center for Applied Scientific Computing

In this work, we analyze two types of summation-by-parts finite difference operators for approximating the second derivative with variable coefficient. The first type uses ghost points, while the second type does not use any ghost points. A previously unexplored relation between the two types of summation-by-parts operators is investigated. By combining them we develop a new fourth order accurate finite difference discretization with hanging nodes on the mesh refinement interface. We take the model problem as the two-dimensional acoustic wave equation in second order form in terms of acoustic pressure, and we prove energy stability for the proposed method. Compared to previous approaches using ghost points, the proposed method leads to a smaller system of linear equations that needs to be solved for the ghost point values. Another attractive feature of the proposed method is that the explicit time step does not need to be reduced relative to the corresponding periodic problem. Numerical experiments, both for smoothly varying and discontinuous material properties, demonstrate that the proposed method converges to fourth order accuracy. A detailed comparison of the accuracy and the time-step restriction with the simultaneous-approximation-term penalty method is also presented. (An erratum is attached.)

• Research Organization: Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
• Sponsoring Organization: USDOE National Nuclear Security Administration (NNSA)
• Grant/Contract Number: AC52-07NA27344
• OSTI ID: 1734602
• Report Number(s): LLNL-JRNL--757334; 945085
• Journal Information: SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 5 Vol. 41; ISSN 1064-8275
• Publisher: SIAM
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
finite difference methods
ghost points
mesh refinement
nonconforming
summation-by-parts
wave equation