A Finite Difference Method with Non-uniform Timesteps for Fractional Diffusion Equations

TL;DR

An implicit finite difference method with non-uniform timesteps for solving the fractional diffusion equation in the Caputo form that is unconditionally stable and convergent.

Abstract

An implicit finite difference method with non-uniform timesteps for solving the fractional diffusion equation in the Caputo form is proposed. The method allows one to build adaptive methods where the size of the timesteps is adjusted to the behaviour of the solution in order to keep the numerical errors small without the penalty of a huge computational cost. The method is unconditionally stable and convergent. In fact, it is shown that consistency and stability implies convergence for a rather general class of fractional finite difference methods to which the present method belongs. The huge computational advantage of adaptive methods against fixed step methods for fractional diffusion equations is illustrated by solving the problem of the dispersion of a flux of subdiffusive particles stemming from a point source.

Figures (3)

• Figure 1: Exact solution $u(x,t)$ (lines) and numerical solutions $U_j^{(n)}$ with variable timesteps given by (\ref{['dtn']}) (symbols) for the problem described in the main text with (from top to bottom at $x=0$) $t=1.0004$ (66 timesteps, down triangles), $t=4.08\times 10^{-4}$ (one timestep, squares), $t=0.034$ (10 timesteps, diamonds), $t=2.0$ (141 timesteps, circles), and $t=1$ (65 timesteps, up triangles). The lines for $t>1$ are dashed. Inset: logarithmic scale and tails of the solutions.
• Figure 2: Exact solution $u(0,t)$ (line) and numerical solution $U_0^{(n)}$ (symbols) at the origin $x=0$ for the problem described in the main text for fixed timesteps with $t_{m+1}-t_m=0.001$ (circles) and for variable timesteps given by (\ref{['dtn']}) (solid squares) in the time interval $0\le t\le 0.1$. Inset: solutions in the whole time interval. In this panel only 1 of every 20 points for the case with fixed timesteps is plotted.
• Figure 3: Absolute difference between the exact solution and the numerical solution at $x=0$, $|e(0,t)|$, for the problem described in the main text for fixed timesteps with $t_{m+1}-t_m =0.001$ (circles) and variable timesteps given by (\ref{['dtn']}) (squares) in the time interval $0\le t\le 0.1$. Inset: error in the whole time interval. In this small panel only 1 of every 20 points for the case with fixed timesteps is plotted.