ON OPTIMAL CELL AVERAGE DECOMPOSITION FOR HIGH-ORDER BOUND-PRESERVING SCHEMES OF HYPERBOLIC CONSERVATION LAWS

Cui, Shumo1; Ding, Shengrong2; Wu, Kailiang3,4,5

2024

10.1137/23M1549365

SIAM JOURNAL ON NUMERICAL ANALYSIS   影响因子和分区

0036-1429

1095-7170

Cell average decomposition (CAD) plays a critical role in constructing boundpreserving (BP) high -order discontinuous Galerkin and finite volume methods for hyperbolic conservation laws. Seeking optimal CAD (OCAD) that attains the mildest BP Courant--Friedrichs--Lewy (CFL) condition is a fundamentally important yet difficult problem. The classic CAD, proposed in 2010 by Zhang and Shu using the Gauss-Lobatto quadrature, has been widely used over the past decade. Zhang and Shu only checked for the 1D \BbbP2 and \BbbP3 spaces that their classic CAD is optimal. However, we recently discovered that the classic CAD is generally not optimal for the multidimensional \BbbP2 and \BbbP3 spaces. Yet, it remained unknown for a decade what CAD is optimal for higher -degree polynomial spaces, especially in multiple dimensions. This paper presents the first systematical analysis and establishes the general theory on the OCAD problem, which lays a foundation for designing more efficient BP schemes. The analysis is very nontrivial and involves novel techniques from several branches of mathematics, including Carathe'\o dory's theorem from convex geometry, and the invariant theory of symmetric group in abstract algebra. Most notably, we discover that the OCAD problem is closely related to polynomial optimization of a positive linear functional on the positive polynomial cone, thereby establishing four useful criteria for examining the optimality of a feasible CAD. Using the established theory, we rigorously prove that the classic CAD is optimal for general 1D \BbbPk spaces and general 2D \BbbQk spaces of an arbitrary k \geq 1. For the widely used 2D \BbbPk spaces, the classic CAD is, however, not optimal, and we develop a generic approach to find out the genuine OCAD and propose a more practical quasi -optimal CAD, both of which provide much milder BP CFL conditions than the classic CAD yet require much fewer nodes. These findings notably improve the efficiency of general high -order BP methods for a large class of hyperbolic equations while requiring only a minor adjustment of the implementation code. The notable advantages in efficiency are further confirmed by numerical results.

hyperbolic conservation laws bound-preserving schemes discontinuous Galerkin methods finite volume methods cell average decomposition

SCI

英语

第一 ; 通讯

Shenzhen Science and Tech-nology Program[RCJC20221008092757098] ; National Natural Science Foundation of China[12171227]

Mathematics

Mathematics, Applied

WOS:001184361800001

SIAM PUBLICATIONS

MATHEMATICS

Web of Science

1.Southern Univ Sci & Technol, SUSTech Int Ctr Math, Shenzhen 518055, Peoples R China
2.Southern Univ Sci & Technol, Dept Math, Shenzhen 518055, Peoples R China
3.Southern Univ Sci & Technol, Dept Math, Shenzhen 518055, Guangdong, Peoples R China
4.Southern Univ Sci & Technol, SUSTech Int Ctr Math, Shenzhen 518055, Guangdong, Peoples R China
5.Natl Ctr Appl Math Shenzhen NCAMS, Shenzhen 518055, Guangdong, Peoples R China

Cui, Shumo,Ding, Shengrong,Wu, Kailiang. ON OPTIMAL CELL AVERAGE DECOMPOSITION FOR HIGH-ORDER BOUND-PRESERVING SCHEMES OF HYPERBOLIC CONSERVATION LAWS[J]. SIAM JOURNAL ON NUMERICAL ANALYSIS,2024,62(2).

Cui, Shumo,Ding, Shengrong,&Wu, Kailiang.(2024).ON OPTIMAL CELL AVERAGE DECOMPOSITION FOR HIGH-ORDER BOUND-PRESERVING SCHEMES OF HYPERBOLIC CONSERVATION LAWS.SIAM JOURNAL ON NUMERICAL ANALYSIS,62(2).

Cui, Shumo,et al."ON OPTIMAL CELL AVERAGE DECOMPOSITION FOR HIGH-ORDER BOUND-PRESERVING SCHEMES OF HYPERBOLIC CONSERVATION LAWS".SIAM JOURNAL ON NUMERICAL ANALYSIS 62.2(2024).