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

摘要

We construct families of approximate solutions to the initial value problemand provide complete mathematical proofs that they tend to satisfy the standardsystem 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 methodthat reduces this system of PDEs to a family of systems of four ODEs in Banachspaces whose smooth solutions are these approximate solutions. This method isconstructive: using standard numerical methods for ODEs one can easily andaccurately compute these approximate solutions which, therefore, from themathematical proof, can serve for comparison with numerical schemes. Oneobserves agreement with previously known solutions from scientific computing[S. Evje, T. Flatten. Hybrid Flux-splitting Schemes for a common two fluidmodel. J. Comput. Physics 192, 2003, p. 175-210]. We show that one recovers thesolutions of these authors (exactly in one case, with a slight difference inanother case). Then we propose an efficient numerical scheme for the originalsystem of two-fluid flows and show it gives back exactly the same results asthe theoretical solutions obtained above.