Optimal long-time decay rate of solutions of complete monotonicity-preserving schemes for nonlinear time-fractional evolutionary equations

TL;DR

This paper establishes the optimal long-time decay rates for solutions of nonlinear time-fractional equations when discretized with complete monotonicity-preserving schemes, extending known continuous decay results to discrete schemes.

Contribution

It proves that $ ext{CM}$-preserving schemes accurately replicate the continuous decay rates for nonlinear fractional equations, including on nonuniform meshes for the L1 scheme.

Findings

Discrete solutions decay as $O(t_{n}^{-rac{\alpha}{\gamma}})$

Results extend to fractional nonlinear subdiffusion problems

Numerical experiments confirm theoretical decay rates

Abstract

The solution of the nonlinear initial-value problem $\mathcal{D}_{t}^{\alpha}y(t)=-\lambda y(t)^{\gamma}$ for $t>0$ with $y(0)>0$, where $\mathcal{D}_{t}^{\alpha}$ is a Caputo derivative of order $\alpha\in (0,1)$ and $\lambda, \gamma$ are positive parameters, is known to exhibit $O(t^{\alpha/\gamma})$ decay as $t\to\infty$. No corresponding result for any discretisation of this problem has previously been proved. In the present paper it is shown that for the class of complete monotonicity-preserving ($\mathcal{CM}$-preserving) schemes (which includes the L1 and Gr\"unwald-Letnikov schemes) on uniform meshes $\{t_n:=nh\}_{n=0}^\infty$, the discrete solution also has $O(t_{n}^{-\alpha/\gamma})$ decay as $t_{n}\to\infty$. This result is then extended to $\mathcal{CM}$-preserving discretisations of certain time-fractional nonlinear subdiffusion problems such as the time-fractional porous media and $p$-Laplace equations. For the L1 scheme, the $O(t_{n}^{-\alpha/\gamma})$ decay result is shown to remain valid on a very general class of nonuniform meshes. Our analysis uses a discrete comparison principle with discrete subsolutions and supersolutions that are carefully constructed to give tight bounds on the discrete solution. Numerical experiments are provided to confirm our theoretical analysis.