@Article{CiCP-24-4,
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 = {<p style="text-align: justify;">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 (MoL<sup>T</sup>) to solve
time-dependent parabolic partial differential equations (PDEs). For the GRK-MoL<sup>T&nbsp;</sup>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-MoL<sup>T&nbsp;</sup>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 &quot;stiffness&quot; in the semi-discrete spatial elliptic differential equations. We
present numerical experiments to validate the theoretical findings.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.2018.hh80.09},
url = {https://global-sci.com/article/79985/on-the-gauss-runge-kutta-and-method-of-lines-transpose-for-initial-boundary-value-parabolic-pdes}
}