Exponential collocation methods for the cubic Schr\"{o}dinger equation

TL;DR

This paper introduces high-order exponential collocation methods for solving the cubic Schrödinger equation on a torus, emphasizing energy preservation and demonstrating superior efficiency through numerical experiments.

Contribution

The paper develops and analyzes new exponential collocation methods that can achieve arbitrarily high order and preserve energy, with detailed proofs and practical implementations.

Findings

Methods can be of arbitrarily high order.

Energy preservation is achieved or nearly achieved.

Numerical experiments show improved efficiency.

Abstract

In this paper we derive and analyse new exponential collocation methods to efficiently solve the cubic Schr\"{o}dinger Cauchy problem on a $d$-dimensional torus. Energy preservation is a key feature of the cubic Schr\"{o}dinger equation. It is proved that the novel methods can be of arbitrarily high order which exactly or nearly preserve the continuous energy of the original continuous system. The existence and uniqueness, regularity, global convergence, nonlinear stability of the new methods are studied in detail. Two practical exponential collocation methods are constructed and two numerical experiments are included. The numerical results illustrate the efficiency of the new methods in comparison with existing numerical methods in the literature.