PATH-CONSERVATIVE CENTRAL-UPWIND SCHEMES FOR NONCONSERVATIVE HYPERBOLIC SYSTEMS

Castro Diaz, Manuel Jesus1; Kurganov, Alexander2,3; Morales de Luna, Tomas4

2019-06-21

10.1051/m2an/2018077

ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE   Impact Factor & Quartile

0764-583X

1290-3841

We develop path-conservative central-upwind schemes for nonconservative one-dimensional hyperbolic systems of nonlinear partial differential equations. Such systems arise in a variety of applications and the most challenging part of their numerical discretization is a robust treatment of nonconservative product terms. Godunov-type central-upwind schemes were developed as an efficient, highly accurate and robust "black-box" solver for hyperbolic systems of conservation and balance laws. They were successfully applied to a large number of hyperbolic systems including several nonconservative ones. To overcome the difficulties related to the presence of nonconservative product terms, several special techniques were proposed. However, none of these techniques was sufficiently robust and thus the applicability of the original central-upwind schemes was rather limited. In this paper, we rewrite the central-upwind schemes in the form of path-conservative schemes. This helps us (i) to show that the main drawback of the original central-upwind approach was the fact that the jump of the nonconservative product terms across cell interfaces has never been taken into account and (ii) to understand how the nonconservative products should be discretized so that their influence on the numerical solution is accurately taken into account. The resulting path-conservative central-upwind scheme is a new robust tool for both conservative and nonconservative hyperbolic systems. We apply the new scheme to the Saint-Venant system with discontinuous bottom topography and two-layer shallow water system. Our numerical results illustrate the good performance of the new path-conservative central-upwind scheme, its robustness and ability to achieve very high resolution.

Nonconservative hyperbolic systems of PDEs Saint-Venant system two-layer shallow water equations central-upwind scheme path-conservative scheme well-balanced scheme

SCI ; EI

English

Corresponding

NSF[DMS-1521009] ; NSF[DMS-1818666]

Mathematics

Mathematics, Applied

WOS:000475769100002

EDP SCIENCES S A

20192807155282

Equations of motion ; Partial differential equations ; Topography

Calculus:921.2 ; Materials Science:951

MATHEMATICS

Web of Science

1.Univ Malaga, Dept Anal Matemat, Malaga 29080, Spain
2.Southern Univ Sci & Technol, Dept Math, Shenzhen 518055, Peoples R China
3.Tulane Univ, Math Dept, New Orleans, LA 70118 USA
4.Univ Cordoba, Dept Matemat, Campus Rabanales, E-14071 Cordoba, Spain

Castro Diaz, Manuel Jesus,Kurganov, Alexander,Morales de Luna, Tomas. PATH-CONSERVATIVE CENTRAL-UPWIND SCHEMES FOR NONCONSERVATIVE HYPERBOLIC SYSTEMS[J]. ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE,2019,53(3):959-985.

Castro Diaz, Manuel Jesus,Kurganov, Alexander,&Morales de Luna, Tomas.(2019).PATH-CONSERVATIVE CENTRAL-UPWIND SCHEMES FOR NONCONSERVATIVE HYPERBOLIC SYSTEMS.ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE,53(3),959-985.

Castro Diaz, Manuel Jesus,et al."PATH-CONSERVATIVE CENTRAL-UPWIND SCHEMES FOR NONCONSERVATIVE HYPERBOLIC SYSTEMS".ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE 53.3(2019):959-985.