Numerical conservation laws of time fractional diffusion PDEs

TL;DR

This paper develops conditions for conservation laws in time fractional diffusion PDEs and proposes a numerical scheme combining finite differences and spectral methods that preserves these laws, validated through numerical experiments.

Contribution

It introduces a general framework for conservation laws in arbitrary order time fractional PDEs and a novel numerical method that preserves these laws.

Findings

The proposed method accurately approximates fractional derivatives of arbitrary order.

Numerical experiments confirm the convergence and conservation properties of the scheme.

The approach extends existing methods to a broader class of fractional diffusion equations.

Abstract

The first part of this paper introduces sufficient conditions to determine conservation laws of diffusion equations of arbitrary fractional order in time. Numerical methods that satisfy a discrete analogue of these conditions have conservation laws that approximate the continuous ones. In the second part of the paper, we propose a method that combines a finite difference method in space with a spectral integrator in time. The time integrator has already been applied in literature to solve time fractional equations with Caputo fractional derivative of order $\alpha\in(0,1)$. It is here generalised to approximate Caputo and Riemann-Liouville fractional derivatives of arbitrary order. We apply the method to subdiffusion and superdiffusion equations with Riemann-Liouville fractional derivative and derive its conservation laws. Finally, we present a range of numerical experiments to show the convergence of the method and its conservation properties.