Fast, Accurate and Robust Adaptive Finite Difference Methods for Fractional Diffusion Equations: The Size of the Timesteps does Matter

TL;DR

Two adaptive methods based on the step-doubling technique are discussed, 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.

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.

Figures (9)

• Figure 1: Scheme of the step-doubling technique. The solution at time $t_n$ is obtained by means of (i) a full timestep of size $t_n-t_{n-1}$ and (ii) by means of two steps of size $(t_n-t_{n-1})/2$. The difference $\mathcal{E}^{(n)}$ between both solutions is used as an indicator of the numerical error.
• Figure 2: Normalized computational time $T_\text{CPU}(t)$ required by the fixed-step method with $\Delta=0.01$ (triangles), by the T&E method (circles), and by the predictive method with $\theta=3/2$ and $\omega=1$ (open squares) and $\omega=1/2$ (solid squares), to solve problem \ref{['testbed']} with $\gamma=1/4$ up to time $t$. In all cases $\Delta x=\pi/40$ and $\tau=10^{-4}$. The lines are guides to the eye; their slopes (0.2 for the dashed line, 0.1 for the dotted line, 2 for the solid line) provide estimates of the power exponent $\beta$ in $T_\text{CPU}(t) \sim t^\beta$.
• Figure 3: Normalized computational time $T_\text{CPU}$ vs $t$ for the method with fixed timesteps with $\Delta=0.01$ (solid triangles) and for the T&E method with tolerance $10^{-5}$ (left triangles), $5\times 10^{-4}$ (circles), $10^{-4}$ (down triangles), $2\times 10^{-4}$ (squares), $10^{-3}$ (open up triangles). In all cases $\gamma=1/4$ and $\Delta x=\pi/40$.
• Figure 4: Normalized computational time $T_\text{CPU}$ vs $t$ when problem \ref{['testbed']} is solved by means of the T&E method with $\tau=10^{-4}$ for $\gamma=0.25, 0.5, 0.75, 0.9, 0.99, 1$ (open circles, squares, up triangles, down triangles, diamonds, stars, respectively) and by the method with fixed timesteps with $\Delta=0.01$ (solid circles). In all cases $\Delta x=\pi/40$.
• Figure 5: Numerical error vs. $t$ when problem \ref{['testbed']} is solved by means of the fixed step method with $\Delta_n =0.01$ (triangles), the T&E method with tolerance $10^{-4}$ (circles) and $10^{-3}$ (squares), and the predictive method with $\omega=1/2$, and tolerance $10^{-4}$ (solid circles) and $10^{-3}$ (solid squares). In all cases $\Delta x=\pi/40$.