06-40
A note on implicit $\theta$-schemes applied on the Navier-Stokes-equations
by Rang, J.
Preprint series: 06-40, Preprints
- MSC:
- 76D05 Navier-Stokes equations, See also {35Q30}
- 35Q30 Stokes and Navier-Stokes equations, See also {76D05, 76D07, 76N10}
- 65L20 Stability of numerical methods
- 65L60 Finite elements, Rayleigh-Ritz, Galerkin and collocation methods
Abstract: In this note second order one-step- and fractional-step-$\theta$-schemes are applied on the semidiscretised Navier-Stokes-equations. Both methods are formulated as Runge-Kutta-methods and are analysed. It is shown that the fractional-step-$\theta$-schemes have only stage order $q=1$ whereas the Crank-Nicolson-scheme has stage order $q=2$. Hence the fractional-step-$\theta$-scheme may have order reduction, if the method is applied on stiff ODEs and DAEs, i.e. the semi-discretised Navier-Stokes equations. Some theoretical results and numerical examples illustrate this phenomena. Moreover it is shown that there exists no fractional-step-$\theta$-method which has the stage order $q=2$ and is strongly A-stable.
Keywords: imcompressible Navier-Stokes equations, implicit $\theta$-schemes, Runge-Kutta-methods, order reduction
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