Solution of the shallow-water equations using an adaptive moving mesh method

摘要

This paper extends an adaptive moving mesh method to multi-dimensional shallow water equations (SWE) with source terms. The algorithm is composed of two independent parts: the SWEs evolution and the mesh redistribution. The first part is a high-resolution kinetic flux-vector splitting (KFVS) method combined with the surface gradient method for initial data reconstruction, and the second part is based on an iteration procedure. In each iteration, meshes are first redistributed by a variational principle and then the underlying numerical solutions are updated by a conservative-interpolation formula on the resulting new mesh. Several test problems in one- and two-dimensions with a general geometry are computed using the proposed moving mesh algorithm. The computations demonstrate that the algorithm is efficient for solving problems with bore waves and their interactions. The solutions with higher resolution can be obtained by using a KFVS scheme for the SWEs with a much smaller number of grid points than the uniform mesh approach, although we do not treat technically the bed slope source terms in order to balance the source terms and flux gradients.