Volume 24, Issue 4
On the Gauss Runge-Kutta and Method of Lines Transpose for Initial-Boundary Value Parabolic PDEs

Bo Zhang, Dangxing Chen & Jingfang Huang

Commun. Comput. Phys., 24 (2018), pp. 1221-1240.

Published online: 2018-06

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  • Abstract

It has been shown in existing analysis that the Gauss Runge-Kutta (GRK) (also called Legendre-Gauss collocation) formulation is super-convergent when applied to the initial value problem of ordinary differential equations (ODEs) in that the discretization error is order 2s when s Gaussian nodes are used. Additionally, the discretized system can be solved accurately and efficiently using the spectral deferred correction (SDC) or Krylov deferred correction (KDC) method. In this paper, we combine the GRK formulation with the Method of Lines Transpose (MoLT) to solve time-dependent parabolic partial differential equations (PDEs). For the GRK-MoLTformulation, we show how the coupled spatial differential equations can be decoupled and efficiently solved using the SDC or KDC method. Preliminary analysis of the GRK-MoLTalgorithm reveals that the super-convergent property of the GRK formulation no longer holds in the PDE case for general boundary conditions, and there exists a new type of “stiffness” in the semi-discrete spatial elliptic differential equations. We present numerical experiments to validate the theoretical findings.

  • Keywords

Gauss Runge-Kutta, Method of Lines Transpose, parabolic system, stiffness, order reduction, spectral deferred correction, Krylov deferred correction.

  • AMS Subject Headings

65F08, 65M12, 65M22, 65M70

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-24-1221, author = {}, title = {On the Gauss Runge-Kutta and Method of Lines Transpose for Initial-Boundary Value Parabolic PDEs}, journal = {Communications in Computational Physics}, year = {2018}, volume = {24}, number = {4}, pages = {1221--1240}, abstract = {

It has been shown in existing analysis that the Gauss Runge-Kutta (GRK) (also called Legendre-Gauss collocation) formulation is super-convergent when applied to the initial value problem of ordinary differential equations (ODEs) in that the discretization error is order 2s when s Gaussian nodes are used. Additionally, the discretized system can be solved accurately and efficiently using the spectral deferred correction (SDC) or Krylov deferred correction (KDC) method. In this paper, we combine the GRK formulation with the Method of Lines Transpose (MoLT) to solve time-dependent parabolic partial differential equations (PDEs). For the GRK-MoLTformulation, we show how the coupled spatial differential equations can be decoupled and efficiently solved using the SDC or KDC method. Preliminary analysis of the GRK-MoLTalgorithm reveals that the super-convergent property of the GRK formulation no longer holds in the PDE case for general boundary conditions, and there exists a new type of “stiffness” in the semi-discrete spatial elliptic differential equations. We present numerical experiments to validate the theoretical findings.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.2018.hh80.09}, url = {http://global-sci.org/intro/article_detail/cicp/12325.html} }
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