An efficient meshless numerical method with the error estimate for two-dimensional Schrödinger equation

Abstract

In this study, we present a capable meshfree numerical approach to numerical solving of the two-dimensional Schrödinger equation (SE). Firstly, the radial point interpolation is obtained to build the shape functions for discretization in the mentioned equation. These meshless shape functions have Kronecker delta attributes. Secondly, the local weak form is constructed over the local sub-domain for the purpose of diminishing the solution of the time-dependent SE in a two-dimensional case and transforming it into a system of algebraic linear equations. Also, semi and fully discrete schemes with their error estimates for this meshless method are obtained. We indicate the proposed method square of error has the order of O(h2m+τ2) in the L2 norm. Moreover, some numerical tests are provided, and their exact and finite difference solutions are compared to verify the efficiency of our algorithm.