Inverse Lax-Wendroff Boundary Treatment for Solving Conservation Laws with Finite Volume Methods

doi:10.1007/s42967-024-00413-x

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (3): 885-909.doi: 10.1007/s42967-024-00413-x

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Inverse Lax-Wendroff Boundary Treatment for Solving Conservation Laws with Finite Volume Methods

Guangyao Zhu¹, Yan Jiang¹, Mengping Zhang¹

School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, Anhui, China

收稿日期:2023-11-30 修回日期:2024-03-30 接受日期:2024-04-14 出版日期:2025-09-20 发布日期:2025-05-23
通讯作者: Yan Jiang, jiangy@ustc.edu.cn;Guangyao Zhu, zhugy@mail.ustc.edu.cn;Mengping Zhang, mpzhang@ustc.edu.cn E-mail:jiangy@ustc.edu.cn;zhugy@mail.ustc.edu.cn;mpzhang@ustc.edu.cn
基金资助:
Research is supported in part by the NSFC Grant 12271499;Research is supported in part by the R&D project of Pazhou Lab (Huangpu) under Grant 2023K0609 and the NSFC Grant 12126604.

Inverse Lax-Wendroff Boundary Treatment for Solving Conservation Laws with Finite Volume Methods

Guangyao Zhu¹, Yan Jiang¹, Mengping Zhang¹

School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, Anhui, China

Received:2023-11-30 Revised:2024-03-30 Accepted:2024-04-14 Online:2025-09-20 Published:2025-05-23
Supported by:
Research is supported in part by the NSFC Grant 12271499;Research is supported in part by the R&D project of Pazhou Lab (Huangpu) under Grant 2023K0609 and the NSFC Grant 12126604.

摘要/Abstract

摘要： In this paper, we concentrate on high-order boundary treatments for finite volume methods solving hyperbolic conservation laws. The complex geometric physical domain is covered by a Cartesian mesh, resulting in the boundary intersecting the grids in various fashions. We propose two approaches to evaluate the cell averages on the ghost cells near the boundary. Both of them start from the inverse Lax-Wendroff (ILW) procedure, in which the normal spatial derivatives at inflow boundaries can be obtained by repeatedly using the governing equations and boundary conditions. After that, we can get an accurate evaluation of the ghost cell average by a Taylor expansion joined with high-order extrapolation, or by a Hermite extrapolation coupling with the cell averages on some “artificial” inner cells. The stability analysis is provided for both schemes, indicating that they can avoid the so-called “small-cell” problem. Moreover, the second method is more efficient under the premise of accuracy and stability. We perform numerical experiments on a collection of examples with the physical boundary not aligned with the grids and with various boundary conditions, indicating the high-order accuracy and efficiency of the proposed schemes.

关键词: Inverse Lax-Wendroff (ILW) method, Numerical boundary treatment, Finite volume method, High-order accuracy, Fixed Cartesian mesh, Hyperbolic conservation laws

Abstract: In this paper, we concentrate on high-order boundary treatments for finite volume methods solving hyperbolic conservation laws. The complex geometric physical domain is covered by a Cartesian mesh, resulting in the boundary intersecting the grids in various fashions. We propose two approaches to evaluate the cell averages on the ghost cells near the boundary. Both of them start from the inverse Lax-Wendroff (ILW) procedure, in which the normal spatial derivatives at inflow boundaries can be obtained by repeatedly using the governing equations and boundary conditions. After that, we can get an accurate evaluation of the ghost cell average by a Taylor expansion joined with high-order extrapolation, or by a Hermite extrapolation coupling with the cell averages on some “artificial” inner cells. The stability analysis is provided for both schemes, indicating that they can avoid the so-called “small-cell” problem. Moreover, the second method is more efficient under the premise of accuracy and stability. We perform numerical experiments on a collection of examples with the physical boundary not aligned with the grids and with various boundary conditions, indicating the high-order accuracy and efficiency of the proposed schemes.

Key words: Inverse Lax-Wendroff (ILW) method, Numerical boundary treatment, Finite volume method, High-order accuracy, Fixed Cartesian mesh, Hyperbolic conservation laws

中图分类号:

65M08
65M12

Guangyao Zhu, Yan Jiang, Mengping Zhang. Inverse Lax-Wendroff Boundary Treatment for Solving Conservation Laws with Finite Volume Methods[J]. Communications on Applied Mathematics and Computation, 2025, 7(3): 885-909.

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Inverse Lax-Wendroff Boundary Treatment for Solving Conservation Laws with Finite Volume Methods

Inverse Lax-Wendroff Boundary Treatment for Solving Conservation Laws with Finite Volume Methods

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