Bound-Preserving Framework for Central-Upwind Schemes for General Hyperbolic Conservation Laws

IF 3 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Scientific Computing Pub Date : 2024-09-10 DOI:10.1137/23m1628024

Shumo Cui, Alexander Kurganov, Kailiang Wu

{"title":"Bound-Preserving Framework for Central-Upwind Schemes for General Hyperbolic Conservation Laws","authors":"Shumo Cui, Alexander Kurganov, Kailiang Wu","doi":"10.1137/23m1628024","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 5, Page A2899-A2924, October 2024. <br/> Abstract. Central-upwind (CU) schemes are Riemann-problem-solver-free finite-volume methods widely applied to a variety of hyperbolic systems of PDEs. Exact solutions of these systems typically satisfy certain bounds, and it is highly desirable and even crucial for the numerical schemes to preserve these bounds. In this paper, we develop and analyze bound-preserving (BP) CU schemes for general hyperbolic systems of conservation laws. Unlike many other Godunov-type methods, CU schemes cannot, in general, be recast as convex combinations of first-order BP schemes. Consequently, standard BP analysis techniques are invalidated. We address these challenges by establishing a novel framework for analyzing the BP property of CU schemes. To this end, we discover that the CU schemes can be decomposed as a convex combination of several intermediate solution states. Thanks to this key finding, the goal of designing BPCU schemes is simplified to the enforcement of four more accessible BP conditions, each of which can be achieved with the help of a minor modification of the CU schemes. We employ the proposed approach to construct provably BPCU schemes for the Euler equations of gas dynamics. The robustness and effectiveness of the BPCU schemes are validated by several demanding numerical examples, including high-speed jet problems, flow past a forward-facing step, and a shock diffraction problem.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"24 1","pages":""},"PeriodicalIF":3.0000,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Scientific Computing","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/23m1628024","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

Abstract

SIAM Journal on Scientific Computing, Volume 46, Issue 5, Page A2899-A2924, October 2024.
Abstract. Central-upwind (CU) schemes are Riemann-problem-solver-free finite-volume methods widely applied to a variety of hyperbolic systems of PDEs. Exact solutions of these systems typically satisfy certain bounds, and it is highly desirable and even crucial for the numerical schemes to preserve these bounds. In this paper, we develop and analyze bound-preserving (BP) CU schemes for general hyperbolic systems of conservation laws. Unlike many other Godunov-type methods, CU schemes cannot, in general, be recast as convex combinations of first-order BP schemes. Consequently, standard BP analysis techniques are invalidated. We address these challenges by establishing a novel framework for analyzing the BP property of CU schemes. To this end, we discover that the CU schemes can be decomposed as a convex combination of several intermediate solution states. Thanks to this key finding, the goal of designing BPCU schemes is simplified to the enforcement of four more accessible BP conditions, each of which can be achieved with the help of a minor modification of the CU schemes. We employ the proposed approach to construct provably BPCU schemes for the Euler equations of gas dynamics. The robustness and effectiveness of the BPCU schemes are validated by several demanding numerical examples, including high-speed jet problems, flow past a forward-facing step, and a shock diffraction problem.

查看原文本刊更多论文

一般双曲守恒定律的中央上风方案的保界框架

SIAM 科学计算期刊》，第 46 卷第 5 期，第 A2899-A2924 页，2024 年 10 月。摘要中央上风（CU）方案是一种无黎曼问题求解器的有限体积方法，广泛应用于各种双曲型 PDE 系统。这些系统的精确解通常满足一定的边界，而数值方案保持这些边界是非常理想的，甚至是至关重要的。在本文中，我们开发并分析了一般双曲守恒律系统的保界（BP）CU 方案。与许多其他戈杜诺夫型方法不同，CU 方案一般不能被重塑为一阶 BP 方案的凸组合。因此，标准的 BP 分析技术就失效了。我们通过建立一个分析 CU 方案 BP 特性的新框架来应对这些挑战。为此，我们发现 CU 方案可以分解为多个中间解状态的凸组合。有了这一关键发现，设计 BPCU 方案的目标就简化为执行四个更易实现的 BP 条件，其中每个条件都可以通过对 CU 方案稍加修改来实现。我们采用所提出的方法为气体动力学欧拉方程构建了可证明的 BPCU 方案。BPCU 方案的稳健性和有效性通过几个苛刻的数值示例得到了验证，包括高速射流问题、流过前向台阶问题和冲击衍射问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

SIAM Journal on Scientific Computing 数学-应用数学

CiteScore

5.50

自引率

3.20%

发文量

209

审稿时长

1 months

期刊介绍： The purpose of SIAM Journal on Scientific Computing (SISC) is to advance computational methods for solving scientific and engineering problems. SISC papers are classified into three categories: 1. Methods and Algorithms for Scientific Computing: Papers in this category may include theoretical analysis, provided that the relevance to applications in science and engineering is demonstrated. They should contain meaningful computational results and theoretical results or strong heuristics supporting the performance of new algorithms. 2. Computational Methods in Science and Engineering: Papers in this section will typically describe novel methodologies for solving a specific problem in computational science or engineering. They should contain enough information about the application to orient other computational scientists but should omit details of interest mainly to the applications specialist. 3. Software and High-Performance Computing: Papers in this category should concern the novel design and development of computational methods and high-quality software, parallel algorithms, high-performance computing issues, new architectures, data analysis, or visualization. The primary focus should be on computational methods that have potentially large impact for an important class of scientific or engineering problems.