An Adaptive Difference Method for Variable-Order Diffusion Equations

TL;DR

This work addresses variable-order fractional-time diffusion equations in Caputo form by developing an adaptive finite-difference method that uses nonuniform timesteps and an L1 discretization of the Caputo derivative. The scheme yields an implicit, unconditionally stable tridiagonal system from a standard three-point spatial discretization, with timesteps chosen by a step-doubling adaptive algorithm to keep local error near a user-defined tolerance. Four VOFPDE cases demonstrate that the adaptive method captures fast short-time dynamics and slow long-time behavior while achieving substantial CPU-time savings over fixed-timestep schemes, even in strongly variable order regimes. The approach generalizes previous constant-order adaptivity to VOFPDEs and offers a flexible framework with potential extensions to other discretizations and adaptive strategies, enabling efficient simulations of heterogeneous diffusion processes.

Abstract

An adaptive finite difference scheme for variable-order fractional-time subdiffusion equations in the Caputo form is studied. The fractional time derivative is discretized by the L1 procedure but using nonhomogeneous timesteps. The size of these timesteps is chosen by an adaptive algorithm in order to keep the local error bounded around a preset value, a value that can be chosen at will. For some types of problems this adaptive method is much faster than the corresponding usual method with fixed timesteps while keeping the local error of the numerical solution around the preset values. These findings turns out to be similar to those found for constant-order fractional diffusion equations.

Figures (8)

• Figure 1: $u(\pi/2, t)$ vs. $t$ for the problem \ref{['eqsCaso1']} of Case 1. The symbols are the numerical solutions provided by the adaptive method with tolerances $\tau=10^{-3}$ (stars), $\tau=5\times 10^{-3}$ (squares) and $\tau=10^{-4}$ (circles). In all cases $\Delta x=\pi/40$. The line is the exact analytical solution. The numerical errors $|U_j^n-u(x_j=1/2,t_n)|$ are plotted in the inset. As reference, we have also included the numerical errors when fixed timesteps of size 0.01 are used (triangles). For the sake of readability, we have only plotted one of every 10 points in this case.
• Figure 2: Normalized computational time $T_\text{CPU}(t)$ required by the fixed-step method with $\Delta_n =0.01$ (triangles) and by the adaptive method with $\tau_1=10^{-3}$ (stars), $\tau_2=5\times 10^{-4}$ (squares) and $\tau_3=10^{-5}$ (circles). They are averaged values over five runs. In all cases $\Delta x=\pi/40$. As reference, we have also plotted two lines corresponding to $T_\text{CPU}\sim t^2$ (solid line) and $T_\text{CPU}\sim t^{1/2}$ (dashed line).
• Figure 3: Adaptive numerical solution $u(x, t)$ of the problem \ref{['eqsCaso2']} (Case 2) with $\Delta x=\pi/40$ and $\tau=10^{-4}$ for times (a) $t=0.0075$, (b) $t=0.507$, (c) $t=3.77$ and (d) $t=1013$ (solid lines). As reference, in panel (a) we have also plotted $u(x,0)$ (dashed line) and the order $\gamma(x)$ of the fractional derivative (dash-dotted line).
• Figure 4: Estimate $|U_j^n-\widetilde{U}_j^n|$ of the error of the adaptive numerical solution $u(x, t)$ of the problem \ref{['eqsCaso2']} (Case 2) when $\Delta x=\pi/40$ and $\tau=10^{-3}$ (stars), $\tau=5\times 10^{-4}$ (squares), and $\tau=10^{-4}$ (circles).
• Figure 5: Adaptive numerical solution $u(x, t)$ of the problem \ref{['eqsCaso3']} (Case 3) with $\Delta x=0.01$ and $\tau=10^{-4}$. The solid lines represent the solution for, from top to bottom, $t=0, 3.8, 9.6, 24.8, 93.0$ and $1350$ (black, green, pink, red, blue, and orange lines, respectively) for $\gamma(x)= (2/5)x(1-x/10)$ (dash-dotted line). The dashed lines are the solutions for the normal diffusion problem (i.e., $\gamma=1$) for, from top to bottom, $t=3.8, 9.6, 24.9$ and $91.8$ (green, pink, red and blue lines, respectively). This last line overlaps the line corresponding to the final stationary solution, which for both $\gamma$ functions is the thin straight line going from $u=1$ to $u=0$.