Stability and Convergence Analysis of Multi-Symplectic Variational Integrator for Nonlinear Schrödinger Equation

Abstract

Stability and convergence analyses of the multi-symplectic variational integrator for the nonlinear Schro¨dinger equation are discussed in this paper. The variational integrator is proved to be unconditionally linearly stable using the von Neumann method. A priori error bound for the scheme is given from the Sobolev inequality and the discrete conservation laws. Subsequently, the variational integrator is derived to converge at O(Δx2+Δt2) in the discrete L2 norm using the energy method. The numerical experimental results match our theoretical derivation.