High-Order Accurate Runge-Kutta (Local) Discontinuous Galerkin Methods for One- and Two-Dimensional Fractional Diffusion Equations

Abstract

As the generalization of the integer order partial differential equations (PDE), the fractional order PDEs are drawing more and more attention for their applications in fluid flow, finance and other areas. This paper presents high-order accurate Runge-Kutta local discontinuous Galerkin (DG) methods for one-and two-dimensional fractional diffusion equations containing derivatives of fractional order in space. The Gaputo derivative is chosen as the representation of spatial derivative, because it may represent the fractional derivative by an integral operator. Some numerical examples show that the convergence orders of the proposed local P'^-DG methods are 0(,h'^'^^) both in one and two dimensions, where P*" denotes the space of the real-valued polynomials with degree at most k.

Local Discontinuous Galerkin methods for fractional diffusion equations

“…Very recently [14] presents a purely qualitative study of the solution of spatial fractional problems in one and two dimensions using a high-order discontinuous Galerkin methods. However, no analysis or theoretical results are offered in this work.…”

A finite difference method for a two-point boundary value problem with a Caputo fractional derivative

“…The problem (1.3) is discussed by Al-Refai (2012a) and is a particular case of the wide class of boundary value problems considered in Pedas and Tamme (2012). It is a steady-state version of the time-dependent problems discussed in Saadatmandi and Dehghan (2011); Shen and Liu (2004/05) ;Zheng et al (2010) and Ji and Tang (2012)-who describe some advantages of the Caputo fractional derivative over the Riemann-Liouville fractional derivative.…”

Nodal discontinuous Galerkin methods for fractional diffusion equations on 2D domain with triangular meshes

“…In 2010, Deng and Hesthaven [9] proposed a local discontinuous Galerkin method for the fractional diffusion equation, and offered stability analysis and error estimates, confirming that the schemes should exhibit optional order of convergence for the superdiffusion case. Almost in the same time, Ji and Tang [13] presented a purely qualitative study of the solution of spatial Caputo fractional problems in one and two dimensions using a high-order Runge-kutta discontinuous Galerkin methods, but did not offer theoretical results.…”

“…In [13], the authors adopted the rectangular meshes to deal with the two-dimensional cases. This paper, as a successor to previous work [9], discusses how to approximate fractional diffusion equations with genuinely unstructured grids beyond one dimension and offer a theoretical analysis for the high dimensional case.…”