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

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

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