arXiv subjects

Mathilde Colombeau

Publications and source records attributed to Mathilde Colombeau.

3 recordsLinked to original sources

arXiv2018

A numerical approximation for the standard one pressure system of two fluid flows with energy equations

We study numerically the standard one pressure model of two fluid flows with energy equations. This system is not solved in time derivative. It has been transformed into an equivalent system solved in time derivative. We show that the scheme in this paper applies to both solved and nonsolved systems and gives same results. One usually adds a nonphysical term to render the system hyperbolic. However, explicit solutions and well posedness of the Cauchy problem for some nonlinear nonhyperbolic systems of physics have been obtained in some events by [B. Keyfitz et al.]. We also show that our scheme applies equally well to both versions, with and without the additional term, whether solved in time derivative or not, which provides four versions of the system. We observe that the nonhyperbolic and the hyperbolic systems give very close but slightly different results: the step values are always the same but peaks in gas and liquid velocities are observed in the nonhyperbolic model, which is typically observed in experimental results concerning the gas kick phenomenon though we are unable to say if this result is related or not. The numerical quality of the hyperbolic solved in time derivative system is better, therefore our results (same pressure and temperatures, and same step values in volume fraction and velocities besides the isolated peaks) provide a justification of the additional term that renders it hyperbolic. Another difficulty lies in that these systems are in nonconservative form and therefore its discontinuous solutions cannot make sense in the theory of distributions and it has been observed that different numerical schemes can lead to different discontinuous solutions. For the hyperbolic system this solution is identical to the main one in [S.T. Munkejord, S. Evje, T. Flatten] obtained from completely different methods.

math.AP↗

arXiv2014

Asymptotic study of the initial value problem to a standard one pressure model of multifluid flows in nondivergence form

We construct families of approximate solutions to the initial value problem and provide complete mathematical proofs that they tend to satisfy the standard system of isothermal one pressure two-fluid flows in 1-D when the data are $L^1$ in densities and $L^\infty$ in velocities. To this end, we use a method that reduces this system of PDEs to a family of systems of four ODEs in Banach spaces whose smooth solutions are these approximate solutions. This method is constructive: using standard numerical methods for ODEs one can easily and accurately compute these approximate solutions which, therefore, from the mathematical proof, can serve for comparison with numerical schemes. One observes agreement with previously known solutions from scientific computing [S. Evje, T. Flatten. Hybrid Flux-splitting Schemes for a common two fluid model. J. Comput. Physics 192, 2003, p. 175-210]. We show that one recovers the solutions of these authors (exactly in one case, with a slight difference in another case). Then we propose an efficient numerical scheme for the original system of two-fluid flows and show it gives back exactly the same results as the theoretical solutions obtained above.

math.AP↗

arXiv2014

Approximate solutions to the initial value problem for some compressible flows in presence of shocks and void regions

For the natural initial conditions $L^1$ in the density field (more generally a positive bounded Radon measure) and $L^\infty$ in the velocity field we obtain global approximate solutions to the Cauchy problem for the 3-D systems of isothermal and isentropic gases, the 2-D shallow water equations and the 3-D system of collisionnal self-gravitating gases. We obtain a sequence of functions which are differentiable in time and continuous in space and tend to satisfy the equations in the sense of distributions in the space variables and in the strong sense in the time variable. The method of construction relies on the study of a specific family of two ODEs in a classical Banach space (one for the continuity equation and one for the Euler equation). Standard convergent numerical methods for the solution of these ODEs can be used to provide concrete approximate solutions. It has been checked in numerous cases in which the solutions of systems of fluid dynamics are known that our constuction always gives back the known solutions. It is also proved it gives the classical analytic solutions in the domain of application of the Cauchy-Kovalevska theorem.

math.AP↗