Michael Sever

Viscous singular shock structure for a nonhyperbolic two-fluid model

We consider a system of two nonhyperbolic conservation laws modelling incompressible two-phase flow in one space dimension. The purpose of this paper is to justify the use of singular shocks in the solution of Riemann problems. We prove that both strictly and weakly overcompressive singular shocks are limits of viscous structures. Using Riemann solutions we solve Cauchy problems with piecewise constant data for the nonhyperbolic two-fluid model.

The Numerical Study of Singular Shocks Regularized by Small Viscosity

A singular shock is a measure valued solution found by considering viscosity limit solutions to certain hyperbolic systems. In this work, some fundamental properties concerning such solutions are derived and then illustrated by extensive numerical study.

Order of Dissipation Near Rarefraction Centers

We consider the approximation of weak solutions of hyperbolic systems of conservation laws,

On the intersection of sets of incoming and outgoing waves

{u_t} + f{(u)_x} = 0, - \infty 0;u( \times,0)given,

Hyperbolic systems of conservation laws with some special invariance properties

(1.1) by projection methods of finite-difference, finite-element, spectral, etc. type (as opposed to methods such as those of Godunov [2] or Glimm [1]). These methods all contain a dissipation term or mechanism, as needed for the generation of entropy in the presence of shocks. (Throughout this discussion, we specialize to systems for which there exists a convex entropy function U, with corresponding entropy flux F [3].) In the presence of shocks, the magnitude of the required dissipation is determined by the requirement that entropy be generated at the correct rate, given a discrete shock profile uniformly bounded (i.e. without excessive overshooting) and confined to a width of 0(h), where h is the mesh size. For example, a regularized form of (1.1) such as

LACK OF HYPERBOLICITY IN THE TWO-FLUID MODEL FOR TWO-PHASE INCOMPRESSIBLE FLOW

{u_t} + f{(u)_x} = h{(A(u,h{u_x}){u_x})_x}