Stability and convergence of a local discontinuous Galerkin method for the fractional diffusion equation with distributed order

Abstract

In this paper, a numerical method is proposed for solving distributed order diffusion equation, which arises in the mathematical modeling of ultra-slow diffusion processes observed in some physical problems, whose solution decays logarithmically as the time t tends to infinity. Based on local discontinuous Galerkin method in space, we develop a fully discrete scheme and prove that the scheme is unconditionally stable and convergent with the order $$O(h^{k+1}+\Delta t+\Delta \alpha ^2)$$ , where $$h, \Delta t$$ , $$\Delta \alpha $$ and k are the step size in space, time, distributed order and the degree of piecewise polynomials, respectively. Extensive numerical examples are carried out to illustrate the effectiveness of the numerical schemes.