Fast Second-Order Evaluation for Variable-Order Caputo Fractional Derivative with Applications to Fractional Sub-Diffusion Equations

Abstract

In this paper, we propose a fast second-order approximation to the variable-order (VO) Caputo fractional derivative, which is developed based on L2-1 σ formula and the exponential-sum-approximation technique. The fast evaluation method can achieve the second-order accuracy and further reduce the computational cost and the acting memory for the VO Caputo fractional derivative. This fast algorithm is applied to construct a relevant fast temporal second-order and spatial fourth-order scheme (F L2-1 σ scheme) for the multi-dimensional VO time-fractional sub-diffusion equations. Theoretically, F L2-1 σ scheme is proved to fulfill the similar properties of the coefficients as those of the well-studied L2-1 σ scheme. Therefore, F L2-1 σ scheme is strictly proved to be unconditionally stable and convergent. A sharp decrease in the computational cost and the acting memory is shown in the numerical examples to demonstrate the efficiency of the proposed method.

A fast linearized virtual element method on graded meshes for nonlinear time-fractional diffusion equations

“…current framework for the convergence analysis of the fast time-stepping method is very simple, which is different from the existing ones [37,40]. It is hopeful that the present framework could simplify the numerical analysis of the fast time-stepping methods for nonlinear timefractional evolution equations.…”

“…The nonlocality of the Caputo fractional derivative operator may make numerical methods for (1.1)-(1.3) very expensive for long time computation. To overcome this difficulty, some fast memory-saving time-stepping methods have been developed to speed up the evaluation of weakly singular kernel [29,[35][36][37][38]. Readers can refer to [29] for more details, where different fast methods are numerically compared, the advantages and disadvantages are displayed.…”

$L2-1_\sigma$ Finite Element Method for Time-Fractional Diffusion Problems with Discontinuous Coefficients

“…Huang et al [14] considered both the L1 discretization and L2-1 σ discretization of the Caputo fractional derivative on graded meshes, analyzed the truncation error, and proved the optimal error bounds in H 1 (Ω). Using an F L2-1 σ scheme, Zhang et al [37] constructed a fast temporal second-order and spatial fourth-order scheme, which greatly reduced the computational cost. Chen and Stynes [5] use Alikhanov's high-order scheme with graded meshes for a time-fractional diffusion problem.…”