On the Gauss Runge-Kutta and Method of Lines Transpose for Initial-Boundary Value Parabolic PDEs (Communications in Computational Physics, 2020)
It has been shown in existing analysis that the Gauss Runge-Kutta (GRK)(also called Legendre-Gauss collocation) formulation is super-convergent when appliedto the initial value problem of ordinary differential equations (ODEs) in thatthe discretization error is order 2s when s Gaussian nodes are used. Additionally,the discretized system can be solved accurately and efficiently using the spectral deferredcorrection (SDC) or Krylov deferred correction (KDC) method. In this paper, wecombine the GRK formulation with the Method of Lines Transpose (MoLT) to solvetime-dependent parabolic partial differential equations (PDEs). For the GRK-MoLT formulation, we show how the coupled spatial differential equations can be decoupledand efficiently solved using the SDC or KDC method. Preliminary analysis of theGRK-MoLT algorithm reveals that the super-convergent property of the GRK formulationno longer holds in the PDE case for general boundary conditions, and there existsa new type of "stiffness" in the semi-discrete spatial elliptic differential equations. Wepresent 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.