High-Order Accurate Runge-Kutta (Local) Discontinuous Galerkin Methods for One- and Two-Dimensional Fractional Diffusion Equations
Year: 2012
Numerical Mathematics: Theory, Methods and Applications, Vol. 5 (2012), Iss. 3 : pp. 333–358
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 Caputo 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 $P^k$-DG methods are $O(h^{k+1})$ both in one and two dimensions, where $P^k$ denotes the space of the real-valued polynomials with degree at most $k$.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.2012.m1107
Numerical Mathematics: Theory, Methods and Applications, Vol. 5 (2012), Iss. 3 : pp. 333–358
Published online: 2012-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 26
Keywords: Discontinuous Galerkin method Runge-Kutta time discretization fractional derivative Caputo derivative diffusion equation.