Error estimates of structure‐preserving Fourier pseudospectral methods for the fractional Schrödinger equation

Abstract

This paper gives a rigorous error analysis of the multisymplectic Fourier pseudospectral method for the nonlinear fractional Schrödinger equation. The method preserves some intrinsic structure properties including the generalized multisymplectic conservation law. By rewriting it in a matrix form similar to that in the finite difference method, the method is shown to be convergent in the discrete L 2 norm with the second-order accuracy in time and spectral accuracy in space. The key techniques in the analysis include the discrete energy method, cutoff of the nonlinearity, and a posterior bound of numerical solutions by using the inverse inequality. In a similar line, the convergence result for the symplectic Fourier pseudospectral method can also be established. Moreover, the errors in the local and global energy conservation laws of discrete systems are also investigated. Numerical tests are performed to confirm the theoretical results. KEYWORDSconservation law, error estimate, Fourier pseudospectral method, fractional Schrödinger equation, structure-preserving method 1 Numer Methods Partial Differential Eq. 2020;36:369-393. wileyonlinelibrary.com/journal/num

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