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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Ernesto Guerrero Fernández & Cipriano Escalante & Manuel J. Castro Díaz, 2021. "Well-Balanced High-Order Discontinuous Galerkin Methods for Systems of Balance Laws," Mathematics, MDPI, vol. 10(1), pages 1-30, December.
    • Handle: RePEc:gam:jmathe:v:10:y:2021:i:1:p:15-:d:707780
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    Cited by:

    1. Carlino, Michele Giuliano & Gaburro, Elena, 2023. "Well balanced finite volume schemes for shallow water equations on manifolds," Applied Mathematics and Computation, Elsevier, vol. 441(C).

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