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On the Gauss Runge-Kutta and Method of Lines Transpose for Initial-Boundary Value Parabolic PDEs

On the Gauss Runge-Kutta and Method of Lines Transpose for Initial-Boundary Value Parabolic PDEs
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Year:    2018

Communications in Computational Physics, Vol. 24 (2018), Iss. 4 : pp. 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-MoLT formulation, 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-MoLT algorithm 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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.2018.hh80.09

Communications in Computational Physics, Vol. 24 (2018), Iss. 4 : pp. 1221–1240

Published online:    2018-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    20

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

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