High-order implicit Galerkin-Legendre spectral method for the two-dimensional Schrodinger equation

TL;DR

This paper introduces a high-order implicit Galerkin-Legendre spectral method combined with Runge-Kutta time integration to efficiently solve the 2D Schrödinger equation with high accuracy and exponential convergence in space.

Contribution

It develops a novel spectral method with proven spectral convergence for the 2D Schrödinger equation, integrating spatial discretization with implicit time-stepping.

Findings

Achieves high-order accuracy in space.

Demonstrates exponential convergence rates.

Effective for complex boundary conditions.

Abstract

In this paper, we propose Galerkin-Legendre spectral method with implicit Runge-Kutta method for solving the unsteady two-dimensional Schrodinger equation with nonhomogeneous Dirichlet boundary conditions and initial condition. We apply a Galerkin-Legendre spectral method for discretizing spatial derivatives, and then employ the implicit Runge-Kutta method for the time integration of the resulting linear first-order system of ordinary differential equations in complex domain. We derive the spectral rate of convergence for the proposed method in the L^2-norm for the semidiscrete formulation. Numerical experiments show our formulation have high-order accurate, and have the exponential rates of convergence in space.