Well-Balanced High-Order Discontinuous Galerkin Methods for Systems of Balance Laws

Ernesto Guerrero Fernández, Cipriano Escalante and Manuel J. Castro Díaz
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Ernesto Guerrero Fernández: Departamento de Análisis Matemático, Estadística e Investigación Operativa y Matemática Aplicada, Facultad de Ciencias, Universidad de Málaga, 29080 Malaga, Spain
Cipriano Escalante: Departamento de Matemáticas, Campus de Rabanales, Universidad de Córdoba, 14071 Cordoba, Spain
Manuel J. Castro Díaz: Departamento de Análisis Matemático, Estadística e Investigación Operativa y Matemática Aplicada, Facultad de Ciencias, Universidad de Málaga, 29080 Malaga, Spain

Mathematics, 2021, vol. 10, issue 1, 1-30

Abstract: This work introduces a general strategy to develop well-balanced high-order Discontinuous Galerkin (DG) numerical schemes for systems of balance laws. The essence of our approach is a local projection step that guarantees the exactly well-balanced character of the resulting numerical method for smooth stationary solutions. The strategy can be adapted to some well-known different time marching DG discretisations. Particularly, in this article, Runge–Kutta DG and ADER DG methods are studied. Additionally, a limiting procedure based on a modified WENO approach is described to deal with the spurious oscillations generated in the presence of non-smooth solutions, keeping the well-balanced properties of the scheme intact. The resulting numerical method is then exactly well-balanced and high-order in space and time for smooth solutions. Finally, some numerical results are depicted using different systems of balance laws to show the performance of the introduced numerical strategy.

Keywords: systems of balance laws; exactly well-balanced; high-order methods; discontinuous Galerkin methods; ADER; Runge–Kutta; WENO; reconstruction operators (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: Add references at CitEc
Citations: View citations in EconPapers (1)

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