Exponential collocation methods for the cubic Schrödinger equation
For researchers solving nonlinear Schrödinger equations, this provides high-order energy-preserving numerical methods, though the improvement is incremental over existing exponential integrators.
The paper derives new exponential collocation methods for the cubic Schrödinger equation on a d-dimensional torus, achieving arbitrarily high order while exactly or nearly preserving energy. Numerical experiments demonstrate improved efficiency over existing methods.
In this paper we derive and analyse new exponential collocation methods to efficiently solve the cubic Schrödinger Cauchy problem on a $d$-dimensional torus. Energy preservation is a key feature of the cubic Schrödinger 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.