### 2006-45

#### Automatic step size selection for the fractional-step-theta-scheme

by Rang, J..

Series: 2006-45, Preprints

MSC:
76D05 ~Navier-Stokes equations
35Q30 ~Navier-Stokes equations
65L20 Stability and convergence of numerical methods
65L60 Finite elements, ~Rayleigh-Ritz, Galerkin and collocation methods

Abstract:
In this note the fractional-step-$\theta$-scheme is written as a diagonally implicit Runge-Kutta method (DIRK method). In this context a strongly A-stable embedded formula of order one is created such that an ODE or a DAE can be solved with automatic step length control. Moreover
the implementation of Runge-Kutta methods applied on DAEs of index 2 is explained. Finally, we apply the new method with automatic step size control on the incompressible Navier-Stokes equations. The numerical results show the advantage of the method.

Keywords:
imcompressible Navier-Stokes equations, implicit $\theta$-schemes, Runge-Kutta-methods, order reduction