RBF-DQ Solution for Shallow Water Equations

Abstract

A mesh-free and highly convergent radial basis function–based differential quadrature method (RBF-DQ) is implemented to simulate shallow water waves in the marine environment and open channels. RBF-DQ is applicable in waterways of regular or irregular geometries for the simulation of currents and tidal waves. The multiquadric (MQ)-type radial basis function is used for the purpose of this study. MQ radial basis function has one or two shape parameters whose values significantly affect the accuracy of the results. To find appropriate values of the shape parameters, a parameter estimation methodology is introduced based on cross validation. The accuracy of RBF-DQ is evaluated by two idealized numerical examples along with a field application for the Oresund Strait located between Sweden and Denmark. In each case, either analytical, numerical solutions from other software programs or measured data were taken as benchmark solutions. Results of this study show that RBF-DQ, unlike conventional polynomial-based DQ, can be applied to irregular domains for tidal wave simulation. The method has two main advantages; first, it is mesh free and does not need mesh generation; second, with many fewer nodes, the results obtained compared well with analytical and benchmark solutions of other numerical schemes.