TY - JOUR T1 - On the Gauss Runge-Kutta and Method of Lines Transpose for Initial-Boundary Value Parabolic PDEs AU - Bo Zhang, Dangxing Chen & Jingfang Huang JO - Communications in Computational Physics VL - 4 SP - 1221 EP - 1240 PY - 2018 DA - 2018/06 SN - 24 DO - http://doi.org/10.4208/cicp.2018.hh80.09 UR - https://global-sci.org/intro/article_detail/cicp/12325.html KW - Gauss Runge-Kutta, Method of Lines Transpose, parabolic system, stiffness, order reduction, spectral deferred correction, Krylov deferred correction. AB -

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.