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An Energy-Stable High-Order Central Difference Scheme for the Two-Dimensional Shallow Water Equations

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Abstract

An energy-stable high-order central finite difference scheme is derived for the two-dimensional shallow water equations. The scheme is mathematically formulated using the semi-discrete energy method for initial boundary value problems described in Olsson (1995, Math. Comput. 64, 1035–1065): after symmetrizing the equations via a change to entropy variables, the flux derivatives are entropy-split enabling the formulation of a semi-discrete energy estimate. We show experimentally that the entropy-splitting improves the stability properties of the fully discretized equations. Thus, the dependence on numerical dissipation to keep the scheme stable for long term time integrations is reduced relative to the original unsplit form, thereby decreasing non-physical damping of solutions. The numerical dissipation term used with the entropy-split equations is in a form which preserves the semi-discrete energy estimate. A random one-dimensional dam break calculation is performed showing that the shock speed is computed correctly for this particular case, however it is an open question whether the correct shock speed will be computed in general

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Author information

Correspondence to Matthew Brown.

Additional information

MSC: 35Q35; 65M12; 65M06

Supported in part by the New Zealand Marsden Fund, grant UOA827

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Brown, M., Gerritsen, M. An Energy-Stable High-Order Central Difference Scheme for the Two-Dimensional Shallow Water Equations. J Sci Comput 28, 1–30 (2006). https://doi.org/10.1007/s10915-005-9005-4

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Keywords

  • Shallow water equations
  • entropy-splitting
  • semi-discrete energy estimate
  • central finite difference scheme