A note on implicit $\theta$-schemes applied on the Navier-Stokes-equations

by Rang, J.


Preprint series: 06-40, Preprints

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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Letzte Änderung: 01.03.2018 - Ansprechpartner:

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