Convergence Analysis of a Finite Volume Scheme for a Distributed Order Diffusion Equation

Abstract

We consider a distributed order diffusion equation with space-dependent conductivity. The distributed order operator is defined via an integral of the usual fractional Caputo derivative multiplied by a weight function $\omega$, i.e. $\displaystyle \mathbb {D}_t^\omega u(t)=\int _0^1 \omega (\alpha )\partial _t^\alpha u(t) d\alpha$, where $\partial _t^\alpha$ is the Caputo derivative of order $\alpha$ given by $\displaystyle \partial _t^\alpha u(t)=\frac{1}{\varGamma (1-\alpha )}\int _0^t(t-s)^{-\alpha }u_s(s)ds$.

We establish a new fully discrete finite volume scheme in which the discretization in space is performed using the finite volume method developed in [9] whereas the discretization of the distributed order operator $\displaystyle \mathbb {D}_t^\omega u$ is given by an approximation of the integral, over the unit interval, using the known Mid Point rule and the approximation of the Caputo derivative $\displaystyle \partial _t^\alpha u$ is defined by the known L1-formula on the uniform temporal mesh.

We prove rigorously new error estimates in $L^\infty (L^2)$ and $L^2(H^1)$–discrete norms. These error estimates are obtained thanks to a new well developed discrete a priori estimate and also to the fact that the full discretization of the distributed-order fractional derivative leads to multi-term fractional order derivatives but the number of these terms is varying accordingly with the approximation of the integral over (0, 1).

This note is a continuation of our previous work [6] which dealt with the Gradient Discretization method (GDM) for time fractional-diffusion equation in which the fractional order derivative is fixed and it is given in the Caputo sense (without consideration the distributed-order fractional derivative) and conductivity is equal to one.

Supported by MCS team (LAGA Laboratory) of the “Université Sorbonne- Paris Nord”.