A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations

• In this work, we design an arbitrary high order accurate nodal discontinuous Galerkin spectral element type method for the one dimensional shallow water equations. The novel method uses a skew-symmetric formulation of the continuous problem. We prove that this discretisation exactly preserves the local mass and momentum. Furthermore, we show that combined with a special numerical interface flux function, the method exactly preserves the entropy, which is also the total energy for the shallow water equations. Finally, we prove that the surface fluxes, the skew-symmetric volume integrals, and the source term are well balanced. Numerical tests are performed to demonstrate the theoretical findings.

Subject headings

NATURVETENSKAP  -- Matematik -- Beräkningsmatematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Computational Mathematics (hsv//eng)

NATURVETENSKAP  -- Geovetenskap och miljövetenskap -- Oceanografi, hydrologi och vattenresurser (hsv//swe)

NATURAL SCIENCES  -- Earth and Related Environmental Sciences -- Oceanography, Hydrology and Water Resources (hsv//eng)

Keyword

skew-symmetric shallow water equations

discontinuous Galerkin spectral element method

Gauss-Lobatto Legendre

summation-by-parts

entropy conservation

well balanced

Publication and Content Type

ref (subject category)

art (subject category)