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Godunov-type methods for conservation laws with a flux function discontinuous in space
Scalar conservation laws with a flux function discontinuous in space are approximated using a Godunov-type method for which a convergence theorem is proved. The case where the flux functions at the interface intersect is emphasized. A very simple formula is given for the interface flux. A numerical comparison between the Godunov numerical flux and the upstream mobility flux is presented for two-phase flow in porous media. A consequence of the convergence theorem is an existence theorem for the solution of the scalar conservation laws under consideration.Furthermore, for regular solutions, uniqueness has been shown
Applications of the DFLU flux to systems of conservation laws
The DFLU numerical flux was introduced in order to solve hyperbolic scalar
conservation laws with a flux function discontinuous in space. We show how this
flux can be used to solve systems of conservation laws. The obtained numerical
flux is very close to a Godunov flux. As an example we consider a system
modeling polymer flooding in oil reservoir engineering
On the upstream mobility scheme for two-phase flow in porous media
When neglecting capillarity, two-phase incompressible flow in porous media is
modelled as a scalar nonlinear hyperbolic conservation law. A change in the
rock type results in a change of the flux function. Discretizing in
one-dimensional with a finite volume method, we investigate two numerical
fluxes, an extension of the Godunov flux and the upstream mobility flux, the
latter being widely used in hydrogeology and petroleum engineering. Then, in
the case of a changing rock type, one can give examples when the upstream
mobility flux does not give the right answer.Comment: A preprint to be published in Computational Geoscience
A Hybrid Godunov Method for Radiation Hydrodynamics
From a mathematical perspective, radiation hydrodynamics can be thought of as
a system of hyperbolic balance laws with dual multiscale behavior (multiscale
behavior associated with the hyperbolic wave speeds as well as multiscale
behavior associated with source term relaxation). With this outlook in mind,
this paper presents a hybrid Godunov method for one-dimensional radiation
hydrodynamics that is uniformly well behaved from the photon free streaming
(hyperbolic) limit through the weak equilibrium diffusion (parabolic) limit and
to the strong equilibrium diffusion (hyperbolic) limit. Moreover, one finds
that the technique preserves certain asymptotic limits. The method incorporates
a backward Euler upwinding scheme for the radiation energy density and flux as
well as a modified Godunov scheme for the material density, momentum density,
and energy density. The backward Euler upwinding scheme is first-order accurate
and uses an implicit HLLE flux function to temporally advance the radiation
components according to the material flow scale. The modified Godunov scheme is
second-order accurate and directly couples stiff source term effects to the
hyperbolic structure of the system of balance laws. This Godunov technique is
composed of a predictor step that is based on Duhamel's principle and a
corrector step that is based on Picard iteration. The Godunov scheme is
explicit on the material flow scale but is unsplit and fully couples matter and
radiation without invoking a diffusion-type approximation for radiation
hydrodynamics. This technique derives from earlier work by Miniati & Colella
2007. Numerical tests demonstrate that the method is stable, robust, and
accurate across various parameter regimes.Comment: accepted for publication in Journal of Computational Physics; 61
pages, 15 figures, 11 table
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