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

TL;DR

This paper develops a mathematical framework to approximate solutions of a one-pressure two-fluid flow model, enabling accurate numerical computations and validation against existing scientific results.

Contribution

It introduces a constructive method reducing PDEs to ODEs in Banach spaces, facilitating precise approximation and comparison with established numerical schemes.

Findings

Approximate solutions converge to the standard two-fluid flow system.

Numerical scheme reproduces theoretical solutions accurately.

Agreement with previous scientific computing results.

Abstract

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.