The space-splitting idea combined with local radial basis function meshless approach to simulate conservation laws equations

Abstract

One acceptable technique in meshfree methods is collocation procedure based on the radial basis functions. But the mentioned technique is poor for solving problems that have shock (advection problems). The use of local collocation radial basis function method overcomes this important issue. The interpolation matrix of local radial basis functions is well-posed. Also, it is better, for obtaining high-order accurate results, we consider many points in the domain; thus, the size of coefficient matrix will be large. We suggest a robust technique to deal with this matter. In the current paper, we combine the local collocation radial basis functions method with the space-splitting technique. Solving some one-dimensional problems instead of a two-dimensional problem is an important advantage of the presented combination. For showing the efficiency of new method, we solve some conservation laws equations. An important benefit of the meshless methods is applying them on irregular domains; thus, we provided a test problem that has been solved on two irregular geometries. We use the local RBF collocation method to discrete the spatial domain and then we obtain a nonlinear system of ODEs that it will be solved using the fourth-order exponential time differencing Runge–Kutta method. Moreover, several examples are given that show the acceptable accuracy and proficiency of the present scheme.