Exponential collocation methods based on continuous finite element approximations for efficiently solving the cubic Schrödinger equation

Abstract

In this paper we derive and analyze new exponential collocation methods to efficiently solve the cubic Schrödinger Cauchy problem on a d-dimensional torus. The novel methods are formulated based on continuous time finite element approximations in a generalized function space. 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 approximately preserve the continuous energy of the original continuous system. The existence and uniqueness, regularity, and convergence of the new methods are studied in detail. Two practical exponential collocation methods are constructed, and three illustrative numerical experiments are included. The numerical results show the remarkable accuracy and efficiency of the new methods in comparison with existing numerical methods in the literature.

Exploiting Kronecker structure in exponential integrators: Fast approximation of the action of φ-functions of matrices via quadrature

“…where n must be replaced with n − 1 if n is odd. The bound ( 28) is (19). In this case, the minimum is attained…”

“…Current approaches include rational Pad approximations [13,14], Krylov subspace methods [15,16], and truncated Taylor series expansion [17]. These new developments led to the application of exponential integrators in a wide range of applications [18,19,20].…”