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Shallow water equations: split-form, entropy stable, well-balanced, and positivity preserving numerical methods. (English) Zbl 1432.65156

Summary: For the first time, a general two-parameter family of entropy conservative numerical fluxes for the shallow water equations is developed and investigated. These are adapted to a varying bottom topography in a well-balanced way, i.e., preserving the lake-at-rest steady state. Furthermore, these fluxes are used to create entropy stable and well-balanced split-form semidiscretisations based on general summation-by-parts (SBP) operators, including Gauß nodes. Moreover, positivity preservation is ensured using the framework of X. Zhang and C.-W. Shu [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 467, No. 2134, 2752–2776 (2011; Zbl 1222.65107)]. Therefore, the new two-parameter family of entropy conservative fluxes is enhanced by dissipation operators and investigated with respect to positivity preservation. Additionally, some known entropy stable and positive numerical fluxes are compared. Furthermore, finite volume subcells adapted to nodal SBP bases with diagonal mass matrix are used. Finally, numerical tests of the proposed schemes are performed and some conclusions are presented.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs

Citations:

Zbl 1222.65107

Software:

HLLE; SymPy; SWASHES; Julia
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Full Text: DOI arXiv

References:

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