SOLVING SHALLOW WATER EQUATIONS BY THE SPARSE POINT REPRESENTATION METHOD

Resumen

In this contribution a finite difference scheme (FDS) over a temporally and spatially adaptive sparse mesh is presented. The sparsity of the mesh is achieved using the Sparse Point Representation method, which is based on an interpolating subdivision scheme taking as indicator for the sparse representation the points with wavelet coefficients higher than a given threshold. Our approach is a method for solving time dependent partial differential equations in general, but in this paper, it is tested solving the shallow water equations, which also to the best of our knowledge constitute a new way to solve such equations. For the numerical simulation, a modified leapfrog finite difference scheme is used on the Sparse Point Representation based sparse mesh. The gain in compression and CPU time with respect to the FDS on a uniform mesh for large size meshes is reported. Regarding other adaptive mesh refinement, accuracy improvement is obtained. These facts demonstrate the efficiency of our proposal.
KEYWORDS: adaptive solution of partial differential equations, subdivision scheme, interpolating wavelet transform, refinement criteria
MSC: 65F50