Numerical analysis of a space-time finite volume element method for the nonlinear Schrödinger equation

Abstract

In this paper, for the first time, we propose a space-time finite volume element (STFVE) method for the cubic nonlinear Schrödinger (NLS) equation. In contrast to the space-time finite element (STFE) method, this method not only is easy to achieve high accuracy in both space and time directions, but also the method itself can maintain the conservation laws of physical quantities, and thus it is well suited to solve the conservation laws equations. For the constructed STFVE scheme, the rigorous theoretical analyses are given including the proof of the existence of the resulting approximations and the optimal $L^{\infty }\left(L^2\right)$ and $L^{\infty }\left(H^1\right)$ norms estimates are obtained in the case that the spatial mesh parameter is not related to the time step size. Finally, some numerical tests are shown to confirm the theoretical findings, unconditional stability and the conservation properties of the STFVE method. Also, the numerical tests show that the STFVE method simulates the NLS equation well.