Extrapolated local radial basis function collocation method for shallow water problems

This paper provides a novel meshfree numerical method – the extrapolated local-radial-basis-function collocation method (ELRBFCM) – to solve the well-known shallow-water equations (or de Saint-Venant equations). To promote the accuracy and stability of the local-radial-basis-function collocation method (LRBFCM), the extrapolation method is employed in the ELRBFCM. The numerical experiments include the simulation of two linear and three non-linear problems. For the linear problems, the results of the extrapolation method not only have good agreement with analytical solutions, but also show better efficiency than the Euler and RK4 scheme. It is convinced that by the extrapolation process, the approximating order and stability of time integration is automatically controlled. For the non-linear problems, the solutions by the proposed method capture the physical reaction that also obtained in the benchmark by a complex high-order finite-volume method. As a result, it is concluded that even for very long period, the meshfree ELRBFCM with high-order temporal approximation can provide robust, high accurate and great efficient numerical simulation for the shallow-water problems.