The Structure of the Riemann Solution for Non-Strictly Hyperbolic Conservation Laws

Summary

We describe the full Riemann solution for a system of two equations which possesses an umbilic point where the characteristic speeds coincide. The solution contains many nontrivial topological features to be expected in non-strictly hyperbolic problems. The model we solve describes the flow of three immiscible fluids in porous media. Despite simplifying assumptions on physical properties of the fluids, the model captures the essential global features of the flow.

The solution is too complicated to be obtained analytically. Rather, we used a program designed for the numerical solution of 2X2 Riemann problems. The program has modules which (1) construct local wave curves by a continuation algorithm, (2) construct non-local wave curves using a global search algorithm, (3) construct boundaries across which the nature of the wave curves change, and (4) verify whether shocks are limits of parabolic viscous profiles.

The difficulties in the solution arise because the Hugoniot curves contain disconnected branches of non-contractible shocks, because the classical coordinate system of Lax for the construction of Riemann solutions does not exist near the umbilic point and because there are shocks with viscous profiles which do not obey Lax’s entropy condition. These non classical shocks have to be considered to insure the existence of the solution. The solution depends continuously on the initial data; numerical evidence indicates that it is unique.