Preprint 2003-071

CFL-Free Numerical Schemes for the Two-Fluid Model

Steinar Evje and Tore Flåtten

Abstract: The main purpose of this paper is to construct an implicit numerical scheme for a two-phase flow model, allowing for violation of the CFL-criterion for all waves. Based on the {\em Mixture Flux} (MF) approach developed in \cite{evj03} we propose both a {\em Weakly Implicit} MF (WIMF) scheme, similar to the one studied in \cite{evj03a}, and a {\em Strongly Implicit} MF (SIMF) scheme. The WIMF scheme is stable for a weak CFL condition which relates time steps to the fluid velocity whereas the SIMF scheme is unconditionally stable, at least for a moving contact discontinuity. Both schemes apply AUSM (advection upstream splitting methods) type of convective fluxes.

The SIMF scheme is obtained by enforcing a stronger implicit coupling between the mass equations than the one used for the WIMF scheme. The resulting scheme allows for sequential updating of the momentum and mass variables on a nonstaggered grid by solving two sparse linear systems. The scheme is conservative in all convective fluxes and consistency between the mass variables and pressure is formally maintained. We present numerical simulations indicating that the CFL-free scheme maintains the good accuracy and stability properties of the WIMF scheme as well as an explicit Roe scheme for small time steps.

Moreover, we demonstrate that the WIMF scheme is able to give an {\em exact} resolution of a {\em moving} contact discontinuity. Explicit schemes cannot possess this property since it closely hang on the fact that the time step can be related to the fluid velocity. This feature of the WIMF scheme explains why it is very accurate for calculation of unsteady two-phase flow phenomena, as was also observed in \cite{evj03a}. The SIMF scheme does not possess the "exact resolution" property of WIMF, however, the ability to take larger time steps can be exploited so that more efficient calculations can be made when accurate resolution of sharp fronts is not essential, e.g. to calculate steady state solutions.

Available as PDF (864 Kbytes).
Steinar Evje, <>
Tore Flåtten, <>
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<> December 9 2003.

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