A Time-Spectral Algorithm for Fractional Wave Problems

Abstract

This paper develops a high-accuracy algorithm for time fractional wave problems, which employs a spectral method in the temporal discretization and a finite element method in the spatial discretization. Moreover, stability and convergence of this algorithm are derived, and numerical experiments are performed, demonstrating the exponential decay in the temporal discretization error provided the solution is sufficiently smooth.

Convergence Analysis of a Petrov–Galerkin Method for Fractional Wave Problems with Nonsmooth Data

“…Jin et al [4] analyzed the G1method and the second-order backward difference method for fractional wave equations, and they obtained the accuracies O(τ ) and O(τ 2 ), respectively. In our previous work [6], a time-spectral method for fractional wave problems was designed, which possesses exponential decay in temporal discretization, under the condition that the solution is smooth enough. Recently, to conquer the singularity in time variable, Li et al [7] presented a space-time finite element method for problem (1), and proved that high-order temporal accuracy can still be achieved if appropriate graded temporal grids are adopted.…”

Optimal error estimation of a time-spectral method for fractional diffusion problems with low regularity data

“…Many numerical methods have been developed in the past a dozen years. Among existing works, four types of temporal discretization are most prevailing, i.e., finite difference methods (L-type schemes) [2,24,36,41], convolution quadrature methods [11,13,60,62], finite element methods [27,28,29,31] and spectral methods [26,35,55,64]. Under certain circumstances, problem (1) has an equivalent form like…”

A space-time finite element method for fractional wave problems

“…These algorithms are easy to implement, but are generally of low temporal accuracy. For the second class of algorithms that use the spectral method to discretize the fractional derivatives, we refer the reader to [13,34,35,36,38,30,12]. These algorithms have high-order accuracy if the solution is sufficiently regular.…”