High-order Mass- and Energy-conserving SAV-Gauss Collocation Finite Element Methods for the Nonlinear Schrödinger Equation

Abstract

A family of arbitrarily high-order fully discrete space-time finite element methods are proposed for the nonlinear Schrödinger equation based on the scalar auxiliary variable formulation, which consists of a Gauss collocation temporal discretization and the finite element spatial discretization. The proposed methods are proved to be well-posed and conserving both mass and energy at the discrete level. An error bound of the form O(h p + τ k+1 ) in the L ∞ (0, T ; H 1 )-norm is established, where h and τ denote the spatial and temporal mesh sizes, respectively, and (p, k) is the degree of the space-time finite elements. Numerical experiments are provided to validate the theoretical results on the convergence rates and conservation properties. The effectiveness of the proposed methods in preserving the shape of a soliton wave is also demonstrated by numerical results.

A Numerical Method Based on Operator Splitting Collocation Scheme for Nonlinear Schrödinger Equation

“…Cui et al [3] developed mass-and energy-preserving exponential Runge-Kutta methods for the nonlinear Schrödinger equation. Feng et al [4] developed the high-order mass-and energy-conserving SAV-Gauss collocation finite element methods for the NLS equation. Wang et al [5] used the two-grid finite element method for the NLS equation and the given superconvergence analysis of the scheme.…”

A mass‐ and energy‐preserving numerical scheme for solving coupled Gross–Pitaevskii equations in high dimensions

“…From the stability and long-time computation point of view, a numerical scheme possessing the conservative properties in the discrete sense is preferable and more popular than the non-conservative ones which may lead to non-physical "blow-up." In fact, more and more interests are attracted in designing and analyzing conservative numerical schemes in solving those partial differential equations inheriting conservative or dissipative properties (see [13,18,29,38,40,42] and references therein). Hence, the purpose of this article lies in constructing and analyzing a mass-and energy-preserving finite difference scheme for the high-dimensional CGPEs.…”

Recent Advances in the Numerical Solution of the Nonlinear Schrödinger Equation

“…2 of 27 the energy or other invariants (see, e.g., [52][53][54][55][56][57][58][59][60][61]) have been investigated: the methods we shall deal with, are exactly placed in this latter setting. The basic approach follows what suggested in [62, page 187], namely by suitably using the method of lines: if the PDEs are of Hamiltonian type, [.…”