Fast, accurate and robust adaptive finite difference methods for fractional diffusion equations

Abstract

The computation time required by standard finite difference methods with fixed timesteps for solving fractional diffusion equations is usually very large because the number of operations required to find the solution scales as the square of the number of timesteps. Besides, the solutions of these problems usually involve markedly different time scales, which leads to quite inhomogeneous numerical errors. A natural way to address these difficulties is by resorting to adaptive numerical methods where the size of the timesteps is chosen according to the behaviour of the solution. A key feature of these methods is then the efficiency of the adaptive algorithm employed to dynamically set the size of every timestep. Here we discuss two adaptive methods based on the step-doubling technique. These methods are, in many cases, immensely faster than the corresponding standard method with fixed timesteps and they allow a tolerance level to be set for the numerical errors that turns out to be a good indicator of the actual errors.

Fast algorithm based on TT-M FE system for space fractional Allen–Cahn equations with smooth and non-smooth solutions

“…Different fast algorithms, which cover the fast computation of time fractional derivative, fast algorithm of nonlinear problem in time, fast calculation of nonlinear problem in space, fast computation of Matrix and so forth, have different acceleration strategies and features. Jiang et al [36] proposed a fast method of the time Caputo fractional derivative, which can reduce the computing time resulted in by the nonlocality of fractional derivative; Liu et al [25], Liu et al [27], and Yin et al [14] considered the fast calculation for time FPDEs based on the Xu' s two-grid FE methods [26], which can reduce the calculating time yielded by the nonlinear term; Zhao et al [30] developed a fast Hermite FE algorithm to improve the computational efficiency of Matrix, and presented a block circulant preconditioner; Yuste and Quintana-Murillo [28] presented the fast and robust adaptive methods with finite difference scheme for the time fractional diffusion equations; Xu et al [22], Wu and Zhou [23] considered the parareal algorithms for solving the linear time fractional ordinary or partial differential equations (FO(P)DEs), respectively; Zeng et al [8] presented a unified stable fast time-stepping method for fractional derivative and integral operators. Recently, Liu et al [9] proposed a fast TT-M FE algorithm for time fractional water wave model, which is developed to deal with time-consuming problem of nonlinear iteration used in the standard nonlinear Galerkin FE method for nonlinear term.…”

A High Order Numerical Method for Solving Nonlinear Fractional Differential Equation with Non-uniform Meshes

“…Li et al [7] considered the error estimates of the rectangle formula, trapezoid formula and the predictor-corrector scheme with non-uniform meshes for solving (1) under the assumption that the solution is sufficiently smooth. Other works for solving fractional differential equations with non-uniform meshes may be found in, for example, [7], [14], [17], [18].…”

Detailed error analysis for a fractional Adams method with graded meshes

“…This excellent approach in [21,22] allows to obtain a (relatively) high convergence order without the otherwise required very unnatural smoothness assumptions on the given solution. Other works for solving fractional differential equations with nonuniform meshes may be found in, for example, [12,19,24,25]. Motivated by the ideas in Diethelm [4] and Stynes et al [22], we will introduce a numerical method for solving (1.1) with the graded meshes and we prove that the optimal convergence order uniformly in t n for the proposed numerical method can be recovered when C 0 D α t y(t), α > 0 behaves as t σ , 0 < σ < 1.…”