Numerical solution methods for implicit Runge-Kutta methods of arbitrary high order

Maya Neytcheva
TDB, IT, UU

Abstract:

In this study we consider the efficient implementation of Implicit Runge-Kutta (IRK) methods to solve large systems of ordinary differential equations, originating from finite element discretization of the heat and similar equations, to be solved on large time intervals.

We advocate the IRK methods, based on Radau (or Gauss-Radau) integration method with polynomials, because they also possess the so-called ''strong A-stability''.

For general IRK methods, all the stages are coupled to each other and cannot be factored into separate systems. We show that we can find a very efficient preconditioned iterative solution method and profit from the high accuracy of IRK and the possibility to use large time steps, to ensure short overall solution time.