A meshless generalized finite difference method for solving shallow water equations with the flux limiter technique

TL;DR: In this article, a meshless stable numerical solver is proposed to solve the non-conservative form of shallow water equations, where discontinuous solutions are allowed to transmit during the simulation, and the upwinding spatial derivatives can be approximated at every node using the half-disk shape of the star and generalized finite difference method.

Abstract: In this study, a novel meshless stable numerical solver is proposed to solve the non-conservative form of shallow water equations. Since they form a hyperbolic system of equations, discontinuous solutions are allowed to transmit during the simulation. The generalized finite difference-split coefficient matrix method, recently proposed, is applied and improved using the flux limiter to eliminate the possible-appearing numerical oscillations. In the proposed scheme, the split-coefficient matrix method is adopted to convert the shallow water equations to the characteristic form. Then, the generalized finite difference method and the second-order Runge-Kutta method are employed for spatial and temporal discretization, respectively. The upwinding spatial derivatives can be approximated at every node using the half-disk shape of the star and generalized finite difference method. Applying the flux limiter technique, the expressions can automatically switch the proper discrete order when facing discontinuous solutions. Although the limiter function required the derivatives of different orders, the generalized finite difference method can solve these necessary expressions using the first- and second-order Tayler series. Several numerical examples are provided to demonstrate the capability of the proposed scheme, and the results are compared with other numerical schemes to show the effectiveness of the proposed generalized finite difference-flux limiter method.