Second-order and nonuniform time-stepping schemes for time fractional evolution equations with time-space dependent coefficients

TL;DR

This paper develops second-order, nonuniform time-stepping schemes for two-dimensional time fractional evolution equations with variable coefficients, addressing challenges posed by initial singularities and variable coefficients.

Contribution

It introduces a novel technique for constructing efficient, stable, second-order schemes with variable time steps for complex time fractional PDEs with general coefficients.

Findings

Unconditionally stable schemes under mild coefficient assumptions

Second-order convergence in discrete H^1-norm demonstrated

Numerical experiments confirm theoretical accuracy and stability

Abstract

The numerical analysis of time fractional evolution equations with the second-order elliptic operator including general time-space dependent variable coefficients is challenging, especially when the classical weak initial singularities are taken into account. In this paper, we introduce a concise technique to construct efficient time-stepping schemes with variable time step sizes for two-dimensional time fractional sub-diffusion and diffusion-wave equations with general time-space dependent variable coefficients. By means of the novel technique, the nonuniform Alikhanov type schemes are constructed and analyzed for the sub-diffusion and diffusion-wave problems. For the diffusion-wave problem, our scheme is constructed by employing the recently established symmetric fractional-order reduction (SFOR) method. The unconditional stability of proposed schemes is rigorously discussed under mild assumptions on variable coefficients and, based on reasonable regularity assumptions and weak time mesh restrictions, the second-order convergence is obtained with respect to discrete $H^1$-norm. Numerical experiments are given to demonstrate the theoretical statements.