A Meshless Pseudospectral Approach Applied to Problems with Weak Discontinuities

In this paper, a meshless pseudospectral method is applied to solve problems possessing weak discontinuities on interfaces. To discretize a differential problem, a global interpolation by radial basis functions is used with the collocation procedure. This leads to obtaining the differentiation matrix for the global approximation of the differential operator from the analyzed equation. Using this matrix, the discretization of the problem is straightforward. To deal with the differential equations with discontinuous coefficients on the interface, the meshless pseudospectral formulation is used with the so-called subdomain approach, where proper continuity conditions are used to obtain accurate results. In the present paper, the differentiation matrix for this method is derived and the choice of a proper value of the shape parameter for radial functions in the context of the subdomain approach is studied. The numerical tests show the effectiveness of the method when using regular or unstructured node distribution. They confirm that the approach preserves well-known advantages of the radial basis function collocation method, i.e., rapid convergence, simplicity of the implementation and extends its usage for problems with weak discontinuity.