Applying the Method of Characteristics and the Meshless Localized Radial Basis Function Collocation Method to Solve Shallow Water Equations

Abstract

This paper proposes an accurate and efficient numerical model by combining the method of characteristics (MOC) and the meshless localized radial basis function collocation method (LRBFCM) to simulate the shallow water flow problems. The shallow water equations (SWEs) are classified into a hyperbolic-type partial differential equations (PDEs) system that easily creates numerically unstable results for the case with discontinuous field values or shock waves. To solve this problem, the SWEs are derived into conservative eigensystem form, and then the MOC is applied to capture the change of conservative variables along the characteristic lines. Specifically, the meshless LRBFCM is used to obtain the field values from the conservative variables; it can ease the complexity of the interpolation procedure on characteristic Lagrangian points and preserve the accuracy in transient problems. For the boundary disposal, a fractional time step skill with the characteristic velocity is considered to determine the boundary requirements. The computational nodes can be generated by the uniform or nonuniform distribution, which reduces the difficulty of node generation to obtain efficient and accurate numerical analysis. Six continuous and discontinuous SWEs benchmark examples are simulated and discussed to verify the proposed model. The excellent agreements with the analytical, experimental, and numerical solutions demonstrate the accuracy and efficiency of the algorithm.