Searcharxiv⌕ Search

arXiv subjects

M. Colombeau

Publications and source records attributed to M. Colombeau.

4 recordsLinked to original sources
arXiv2019

The Cauchy problem for the standard one pressure system of two fluid flows with energy equations

We construct with full rigorous mathematical proof a family of approximate solutions to the Cauchy problem for the standard system of two fluid flows with energy equations and we pass to the limit by weak compactness to obtain Radon measures that satisfy the equations in a natural weak sense. Our method provides a convergent numerical method for the numerical calculation of these Radon measures by reducing the system of partial differential equations in the case of these approximate solutions to a system of ordinary differential equations. We observe numerically on the standard Toumi shock tube problem that the Radon measures from our method agree with the numerical solutions previously obtained by other authors with various different numerical methods. In a subsequent numerical paper, using a standard confident scheme with splittings and vanishing viscosity (independent on the above construction), we observe exactly the numerical solution given by our mathematical proof.

math.AP↗
arXiv2011

A numerical scheme for singular shock solutions and a study of its consistence in the sense of distributions

In this paper we present a numerical scheme for the approximation of singular shock solutions of the Keyfitz-Kranzer model system. Consistence in the sense of distributions is studied. As long as some numerical properties are verified when the space step tends to 0, we prove that the scheme provides a numerical solution that satisfies the equations in the sense of distributions with an approximation that tends to 0 when h \rightarrow 0. We also show that this scheme adapts to degenerate systems. This is illustrated by two examples: the system presenting delta wave solutions originally studied by Korchinski and another system studied by Keyfitz-Kranzer that models elasticity. Consistence of the scheme in the sense of distributions is fully proved in the case of the Korchinski model.

math.NA↗
arXiv2009

Numerical approximation of systems modeling large structure formation in cosmology

Numerical approximations of two classical fluid dynamics systems modelling large structure formation in cosmology are proposed. These systems model nonrelativistic and relativistic fluids submitted to self-gravitation in an expanding background. They are obtained by an adaptation of an extension of the Godunov method, using delta wave projections, which was first introduced in Le Roux & al. for the system of pressureless fluid dynamics and which adapts to various systems of fluid dynamics.

math.NA↗