A staggered semi-implicit spectral discontinuous Galerkin scheme for the shallow water equations

► Semi-implicit DG scheme for the shallow water equations on staggered meshes. ► Large time steps, efficient for steady or low Froude number flows. ► Block-penta-diagonal system for only one scalar unknown (free surface). ► Quadrature-free method using precomputed universal matrices and tensors. ► S...

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Bibliographic Details
Published in:Applied mathematics and computation Vol. 219; no. 15; pp. 8057 - 8077
Main Authors: Dumbser, Michael, Casulli, Vincenzo
Format: Journal Article
Language:English
Published: Elsevier Inc 01.04.2013
Subjects:
Algorithms
Discontinuous Galerkin
Free surface flow
Galerkin methods
High order
Large time steps
Mathematical analysis
Mathematical models
Matrices
Semi-implicit
Shallow water equations
Spectra
Spectral
Staggered grid
Wave equations
Staggered grid
Shallow water equations
Discontinuous Galerkin
Large time steps
High order
Semi-implicit
Free surface flow
Spectral
ISSN:0096-3003, 1873-5649
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Summary:► Semi-implicit DG scheme for the shallow water equations on staggered meshes. ► Large time steps, efficient for steady or low Froude number flows. ► Block-penta-diagonal system for only one scalar unknown (free surface). ► Quadrature-free method using precomputed universal matrices and tensors. ► Stability of the method is independent from the gravity wave celerity. A spatially arbitrary high order, semi-implicit spectral discontinuous Galerkin (DG) scheme for the numerical solution of the shallow water equations on staggered control volumes is derived and discussed. The free surface elevation and the momentum are expressed in terms of the same polynomials that are used as basis, and as test functions. Each unknown is, however, defined on a different set of control volumes that are spatially staggered with respect to each other. A semi-implicit time integration yields a stable and efficient mass conservative algorithm. The use of a staggered mesh has the advantage that after substitution of the momentum equations into the mass conservation equation only one single block-penta-diagonal system must be solved for the new free surface location, which is a scalar quantity. Subsequently the new momentum components are directly obtained. The staggered semi-implicit approach makes the present DG scheme different from other published DG schemes. The sparse linear system of the resulting discrete wave equation for the free surface can be conveniently solved by a matrix-free GMRES algorithm. Furthermore, for the shallow water equations the proposed scheme can be written as a quadrature-free method, where all surface and volume integrals can be precomputed once and for all on the reference element and assembled into universal matrices and tensors. For the special case N=0 the proposed method reduces to a classical semi-implicit finite difference scheme. The proposed semi-implicit scheme is particularly well suited for low Froude number flows. The method is validated on some typical academic benchmark problems, using polynomial degrees of up to N=20.
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ISSN:0096-3003
1873-5649
DOI:10.1016/j.amc.2013.02.041