A Weighted Compact Finite Volume Scheme for Hyperbolic Conservation Laws

doi:10.1007/s42967-025-00496-0

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (4): 1243-1267.doi: 10.1007/s42967-025-00496-0

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A Weighted Compact Finite Volume Scheme for Hyperbolic Conservation Laws

Yan Guo¹, Yufeng Shi², Bulong He¹

1. School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, Jiangsu, China;
2. Advanced Analysis & Computation Center, China University of Mining and Technology, Xuzhou, 221116, Jiangsu, China

收稿日期:2024-12-23 修回日期:2025-03-19 接受日期:2025-03-27 出版日期:2026-08-20 发布日期:2026-09-17
通讯作者: Yufeng Shi, Email: shiyufeng@cumt.edu.cn E-mail:shiyufeng@cumt.edu.cn
基金资助:
This work is supported by the Fundamental Research Funds for the Central Universities (2020QN01).

A Weighted Compact Finite Volume Scheme for Hyperbolic Conservation Laws

Yan Guo¹, Yufeng Shi², Bulong He¹

1. School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, Jiangsu, China;
2. Advanced Analysis & Computation Center, China University of Mining and Technology, Xuzhou, 221116, Jiangsu, China

Received:2024-12-23 Revised:2025-03-19 Accepted:2025-03-27 Online:2026-08-20 Published:2026-09-17
Supported by:
This work is supported by the Fundamental Research Funds for the Central Universities (2020QN01).

摘要/Abstract

摘要： In this paper, we introduce a novel fifth-order weighted compact finite volume scheme that integrates the concept of adaptive order weighted essentially non-oscillatory (WENO) schemes into existing linear compact schemes. Our proposed method employs a new stencil selection criterion for the smoothness indicator, resulting in a convex combination of one fifth-order linear compact reconstruction and two third-order linear reconstructions. This innovative combination preserves the high-resolution characteristics of compact schemes while effectively capturing discontinuities with the accuracy of WENO methods. To demonstrate the efficiency of our scheme, we applied it to solve a range of nonlinear scalar equations and systems of Euler equations in both one- and two-dimensional spaces. The results show that the intended fifth-order accuracy is maintained in smooth regions, and the resolution near shocks or discontinuities is significantly enhanced without any numerical oscillations. Additionally, the computational cost associated with using the same stencils is reduced.

关键词: Hyperbolic conservation laws, Compact schemes, Weighted essentially non-oscillatory (WENO), High resolution, Adaptive order

Abstract: In this paper, we introduce a novel fifth-order weighted compact finite volume scheme that integrates the concept of adaptive order weighted essentially non-oscillatory (WENO) schemes into existing linear compact schemes. Our proposed method employs a new stencil selection criterion for the smoothness indicator, resulting in a convex combination of one fifth-order linear compact reconstruction and two third-order linear reconstructions. This innovative combination preserves the high-resolution characteristics of compact schemes while effectively capturing discontinuities with the accuracy of WENO methods. To demonstrate the efficiency of our scheme, we applied it to solve a range of nonlinear scalar equations and systems of Euler equations in both one- and two-dimensional spaces. The results show that the intended fifth-order accuracy is maintained in smooth regions, and the resolution near shocks or discontinuities is significantly enhanced without any numerical oscillations. Additionally, the computational cost associated with using the same stencils is reduced.

Key words: Hyperbolic conservation laws, Compact schemes, Weighted essentially non-oscillatory (WENO), High resolution, Adaptive order

Yan Guo, Yufeng Shi, Bulong He. A Weighted Compact Finite Volume Scheme for Hyperbolic Conservation Laws[J]. Communications on Applied Mathematics and Computation, 2026, 8(4): 1243-1267.

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A Weighted Compact Finite Volume Scheme for Hyperbolic Conservation Laws

A Weighted Compact Finite Volume Scheme for Hyperbolic Conservation Laws

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