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Well-balanced, energy stable schemes for the shallow water equations with varying topography

by U. S. Fjordholm and S. Mishra and E. Tadmor

(Report number 2010-26)

Abstract
We consider the shallow water equations with bottom topography. The smooth solutions of these equations are energy conservative, whereas weak solutions are energy stable. The equations possess interesting steady states like the lake at rest in both one and two space dimensions, as well as moving equilibrium states in one dimension. We design an energy conservative finite volume scheme that preserves the lake at rest as well as moving equilibrium states. Suitable energy stable numerical diusion operators based on energy and equilibrium variables are designed to preserve the lake at rest and moving equilibrium states, respectively. Several numerical experiments illustrating the robustness of the energy preserving and energy stable well-balanced schemes are presented.

Keywords:

@Techreport{FMT10_430,
  author = {U. S. Fjordholm and S. Mishra and E. Tadmor},
  title = {Well-balanced, energy stable schemes for the shallow water equations with varying topography},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2010-26},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2010/2010-26.pdf },
  year = {2010}
}

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