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

Abstract

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.)

Authors:
 [1];  [2]
  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
Publication Date:
Research Org.:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1734602
Report Number(s):
LLNL-JRNL-757334
Journal ID: ISSN 1064-8275; 945085
Grant/Contract Number:  
AC52-07NA27344
Resource Type:
Accepted Manuscript
Journal Name:
SIAM Journal on Scientific Computing
Additional Journal Information:
Journal Volume: 41; Journal Issue: 5; Journal ID: ISSN 1064-8275
Publisher:
SIAM
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; wave equation; finite difference methods; summation-by-parts; ghost points; nonconforming; mesh refinement

Citation Formats

Wang, Siyang, and Petersson, N. Anders. Fourth Order Finite Difference Methods for the Wave Equation with Mesh Refinement Interfaces. United States: N. p., 2019. Web. doi:10.1137/18m1211465.
Wang, Siyang, & Petersson, N. Anders. Fourth Order Finite Difference Methods for the Wave Equation with Mesh Refinement Interfaces. United States. https://doi.org/10.1137/18m1211465
Wang, Siyang, and Petersson, N. Anders. Tue . "Fourth Order Finite Difference Methods for the Wave Equation with Mesh Refinement Interfaces". United States. https://doi.org/10.1137/18m1211465. https://www.osti.gov/servlets/purl/1734602.
@article{osti_1734602,
title = {Fourth Order Finite Difference Methods for the Wave Equation with Mesh Refinement Interfaces},
author = {Wang, Siyang and Petersson, N. Anders},
abstractNote = {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.)},
doi = {10.1137/18m1211465},
journal = {SIAM Journal on Scientific Computing},
number = 5,
volume = 41,
place = {United States},
year = {2019},
month = {10}
}

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