Haar wavelets multi-resolution collocation procedures for two-dimensional nonlinear Schrödinger equation

In this article, two different multi-resolution methods based on Haar wavelets are proposed for the numerical solution of two-dimensional Schrödinger equations. Both the linear and nonlinear model equations are considered. The nonlinear term present in the model equation are linearized by a simple linearization technique. After linearization, the first order finite-difference approximation is used to discretize the time derivative and the Schrödinger equation is then converted into full algebraic form, once the space derivatives are approximated by finite Haar series. The stability analysis of the proposed methods are also studied. Both the proposed methods are time efficient as compared to other methods reported in the literature. To check the accuracy and capabilities of the proposed methods, several benchmark cases are considered as test problems.