Analysis of the L1 scheme for fractional wave equations with nonsmooth data

Abstract

This paper analyzes the well-known L1 scheme for fractional wave equations with nonsmooth data. A new stability estimate is obtained, and the temporal accuracy O(τ 3−α ) is derived for the nonsmooth initial data. In addition, a modified L1 scheme is proposed, and stability and temporal accuracy O(τ 2 ) are derived for this scheme with nonsmooth initial data. The convergence of the two schemes in the inhomogeneous case is also established. Finally, numerical experiments are performed to verify the theoretical results.

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

“…To improve the temporal accuracy, graded meshes were used in [24,49,58] and some correction techniques were proposed in [12,22,30,61]. However, most of the existing works using graded meshes require some assumption of growth estimate on the true solution, and the analyses of correction schemes for (3) are mainly based on the Laplace transform, which is only applicable for uniform temporal grids, and the obtained convergence rates have the form t −q j τ p with 0 < q p (like (3)), which deteriorate near the origin.…”

A symmetric fractional-order reduction method for direct nonuniform approximations of semilinear diffusion-wave equations

“…Lately, for the problem with nonsmooth data, a Petrov-Galerkin method and a time-stepping discontinuous Galerkin method are proposed in [22] (Luo, Li and Xie) and [14] (Li, Wang and Xie), where the temporal convergence rate is (3 − α)/2-order and about first-order respectively. Numerical schemes with classical L1 approximation in time and the standard P1-element in space are also implemented in [13] to have the temporal accuracies of O(τ 3−α ) and O(τ 2 ) provided the ratio τ α /h 2 min is uniformly bounded. We note that the numerical methods in the above works [10,11,13,14,22] are implemented on uniform temporal steps.…”

“…Numerical schemes with classical L1 approximation in time and the standard P1-element in space are also implemented in [13] to have the temporal accuracies of O(τ 3−α ) and O(τ 2 ) provided the ratio τ α /h 2 min is uniformly bounded. We note that the numerical methods in the above works [10,11,13,14,22] are implemented on uniform temporal steps. On the other hand, Mustapha & McLean [29] and Mustapha & Schötzau [30] considered the time-stepping discontinuous Galerkin methods on nonuniform temporal meshes to solve the following kind of fractional wave equation: u t + I β Au(t) = f (t), for β ∈ (0, 1) and t ∈ (0, T ], (1.2) where A is a self-adjoint linear elliptic spatial operator.…”