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Volume 5, Issue 3
High-Order Accurate Runge-Kutta (Local) Discontinuous Galerkin Methods for One- and Two-Dimensional Fractional Diffusion Equations

Xia Ji & Huazhong Tang

DOI: 10.4208/nmtma.2012.m1107

Numer. Math. Theor. Meth. Appl., 5 (2012), pp. 333-358.

Published online: 2012-05

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  • 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 $k$.

  • Keywords

Discontinuous Galerkin method, Runge-Kutta time discretization, fractional derivative, Caputo derivative, diffusion equation.

  • AMS Subject Headings

35R11, 65M60, 65M12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-5-333, author = {Xia Ji and Huazhong Tang}, title = {High-Order Accurate Runge-Kutta (Local) Discontinuous Galerkin Methods for One- and Two-Dimensional Fractional Diffusion Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2012}, volume = {5}, number = {3}, pages = {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 $k$.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2012.m1107}, url = {http://global-sci.org/intro/article_detail/nmtma/5941.html} }
TY - JOUR T1 - High-Order Accurate Runge-Kutta (Local) Discontinuous Galerkin Methods for One- and Two-Dimensional Fractional Diffusion Equations AU - Xia Ji & Huazhong Tang JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 333 EP - 358 PY - 2012 DA - 2012/05 SN - 5 DO - http://doi.org/10.4208/nmtma.2012.m1107 UR - https://global-sci.org/intro/article_detail/nmtma/5941.html KW - Discontinuous Galerkin method, Runge-Kutta time discretization, fractional derivative, Caputo derivative, diffusion equation. AB -

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 $k$.

Xia Ji and Huazhong Tang. (2012). High-Order Accurate Runge-Kutta (Local) Discontinuous Galerkin Methods for One- and Two-Dimensional Fractional Diffusion Equations. Numerical Mathematics: Theory, Methods and Applications. 5 (3). 333-358. doi:10.4208/nmtma.2012.m1107
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