Order-Optimal Preconditioners for Implicit Runge-Kutta Schemes Applied to Parabolic PDEs

In this paper we show that standard preconditioners for parabolic PDEs discretized by implicit Euler or Crank-Nicolson schemes can be reused for higher-order fully implicit Runge-Kutta time discretization schemes. We prove that the suggested block diagonal preconditioners are order-optimal for A-stable and irreducible Runge-Kutta schemes with invertible coefficient matrices. The theoretical investigations are confirmed by numerical experiments.

[1]  Stefan Vandewalle,et al.  Multigrid Methods for Implicit Runge-Kutta and Boundary Value Method Discretizations of Parabolic PDEs , 2005, SIAM J. Sci. Comput..

[2]  Ragnar Winther,et al.  A domain embedding preconditioner for the Lagrange multiplier system , 2000, Math. Comput..

[3]  D. Arnold,et al.  Preconditioning discrete approximations of the Reissner-Mindlin plate model , 1997 .

[4]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[5]  T. Katsaounis.  Advanced Topics in Computational Partial Differential Equations: Numerical Methods and Diffpack Programming , 2005 .

[6]  Uri M. Ascher,et al.  Computer methods for ordinary differential equations and differential-algebraic equations , 1998 .

[7]  H. Hu.  A Projective Method for Rescaling a Diagonally Stable Matrix to be Positive Definite , 1992, SIAM J. Matrix Anal. Appl..

[8]  Francesca Mazzia,et al.  Solving ordinary differential equations by generalized Adams methods: properties and implementation techniques , 1998 .

[9]  Laurent O. Jay,et al.  A parallelizable preconditioner for the iterative solution of implicit Runge-Kutta-type methods , 1999 .

[10]  Bertil Gustafsson,et al.  Deferred Correction Methods for Initial Value Problems , 2001 .

[11]  Jöran Bergh,et al.  General Properties of Interpolation Spaces , 1976 .

[12]  Kent-André Mardal,et al.  Uniform preconditioners for the time dependent Stokes problem , 2004, Numerische Mathematik.

[13]  Hans Petter Langtangen,et al.  Computational Partial Differential Equations - Numerical Methods and Diffpack Programming , 1999, Lecture Notes in Computational Science and Engineering.

[14]  Trygve K. Nilssen,et al.  Preconditioning of fully implicit Runge-Kutta schemes for parabolic PDEs , 2006 .

[15]  Laurent O. Jay,et al.  Inexact Simplified Newton Iterations for Implicit Runge-Kutta Methods , 2000, SIAM J. Numer. Anal..

[16]  T. Nilssen,et al.  Weakly positive definite matrices , 2005 .

[17]  James H. Bramble,et al.  Efficient Higher Order Single Step Methods for Parabolic Problems. Part I. , 1980 .

[18]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[19]  Kent-André Mardal,et al.  Reuse of standard preconditioners for higher-order time discretizations of parabolic PDEs , 2006, J. Num. Math..