Efficient implementation of high order inverse Lax-Wendroff boundary treatment for conservation laws

Sirui Tan Brown University Cheng Wang Beijing Institute of Technology Chi-Wang Shu Brown University Jianguo Ning Beijing Institute of Technology

Numerical Analysis and Scientific Computing mathscidoc:1610.25076

Journal of Computational Physics, 231, 2510-2527, 2012
In earlier work, two of the authors developed a high order accurate numerical boundary condition procedure for hyperbolic conservation laws, which allows the computation using high order finite difference schemes on Cartesian meshes to solve problems in arbitrary physical domains whose boundaries do not coincide with grid lines. This procedure is based on the so-called inverse Lax-Wendroff (ILW) procedure for inflow boundary conditions and high order extrapolation for outflow boundary conditions. However, the algebra of the ILW procedure is quite heavy for two dimensional (2D) hyperbolic systems, which makes it difficult to implement the procedure for order of accuracy higher than three. In this paper, we first discuss a simplified and improved implementation for this procedure, which uses the relatively complicated ILW procedure only for the evaluation of the first order normal derivatives. Fifth order WENO type extrapolation is used for all other derivatives, regardless of the direction of the local characteristics and the smoothness of the solution. This makes the implementation of a fifth order boundary treatment practical for 2D systems with source terms. For no-penetration boundary condition of compressible inviscid flows, a further simplification is discussed, in which the evaluation of the tangential derivatives involved in the ILW procedure is avoided. We test our simplified and improved boundary treatment for Euler equations with or without source terms representing chemical reactions in detonations. The results demonstrate the designed fifth order accuracy, stability, and good performance for problems involving complicated interactions between detonation/shock waves and solid boundaries.
numerical boundary conditions; hyperbolic conservation laws; Cartesian mesh; inverse Lax-Wendroff procedure; reactive Euler equations; detonation; no-penetration conditions
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  title={ Efficient implementation of high order inverse Lax-Wendroff boundary treatment for conservation laws},
  author={Sirui Tan, Cheng Wang, Chi-Wang Shu, and Jianguo Ning},
  booktitle={Journal of Computational Physics},
Sirui Tan, Cheng Wang, Chi-Wang Shu, and Jianguo Ning. Efficient implementation of high order inverse Lax-Wendroff boundary treatment for conservation laws. 2012. Vol. 231. In Journal of Computational Physics. pp.2510-2527. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161012112720235460194.
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