Fast structure-preserving difference algorithm for 2D nonlinear space-fractional wave models

Abstract

In this article, we present an efficient structure-preserving difference scheme for two-dimensional nonlinear space fractional wave models. We use the Crank-Nicolson scheme to discretize in the time direction, and adopt a shifted convolution quadrature (SCQ) formula to approximate the spatial direction. We show and prove an important lemma, and then based on this lemma, we rigorously derive the energy dissipation for the damped case and the approximate energy conservation in undamped case for the fully discrete scheme. In addition, we do the detailed analysis for error estimates with the optimal convergence result O ( h x 2 + h y 2 + τ 2 ) , where h x , h y and τ denote the step length in space-time directions. Furthermore, we provide an efficient fast algorithm called the preconditioned fast BiCG (PF-BiCG) to reduce the cost of time and save memory. Finally, we carry out numerical tests by taking three numerical examples to validate the theories, to observe the performance behaviors of wave solutions, and to reveal the relationship between models by changing fractional parameters.