Numerical Solution of Time-Dependent Schrödinger Equation in 2D Using Method of Particular Solutions with Polynomial Basis Functions

The method of particular solutions using polynomial basis functions (MPS-PBF) has been extensively used to solve various types of partial differential equations. Traditional methods employing radial basis functions (RBFs)—such as Gaussian, multiquadric, and Matérn functions—often suffer from accuracy issues due to their dependence on a shape parameter, which is very difficult to select optimally. In this study, we adopt the MPS-PBF to solve the time-dependent Schrödinger equation in two dimensions. By utilizing polynomial basis functions, our approach eliminates the need to determine a shape parameter, thereby simplifying the solution process. Spatial discretization is performed using the MPS-PBF, while temporal discretization is handled via the backward Euler and Crank–Nicolson methods. To address the ill conditioning of the resulting system matrix, we incorporate a multi-scale technique. To validate the efficacy of the proposed scheme, we present four numerical examples and compare the results with known analytical solutions, demonstrating the accuracy and robustness of the scheme.