Loss of hyperbolicity changes the number of wave groups in Riemann problems

Abstract

Abstract.The main goal of our work is to show that there exists a class of 2×2 Riemann problems for which the solution comprises a single wave group for an open set of initial conditions. This wave group comprises a 1-rarefaction joined to a 2-rarefaction, not by an intermediate state, but by a doubly characteristic shock, 1-left and 2-right characteristic. In order to ensure that perturbations of initial conditions do not destroy the adjacency of the waves, local transversality between a composite curve foliation and a rarefaction curve foliation is necessary.

Resonance in rarefaction and shock curves: Local analysis and numerics of the continuation method

“…In the Riemann solver, we find a Hugoniot curve using two methods. The first is a direct search for points satisfying condition (32), by means of sign changes of the functions in an appropriate grid. The second consists of the continuation method, which is useful in the construction of wave curve.…”

“…However, the Riemann solver allows a major degree of generalization such that the algorithms can be extended to solve the Riemann problem for any system of conservation laws. Some examples using this program are [30,32,31] This exact Riemann solver is useful as a validation tool of the numerical schemes and vise versa. In this sense, a numerical model based on finite difference schemes was incorporated which enables a cross validation.…”

Compositional flow in porous media: Riemann problem for three alkanes

“…Another restriction is that the temperature must stay away from critical value. Our problem differs from the compositional problems studied in [3,6,17] since in our case the volume changes with composition variation.…”

“…Our problem differs from the problems in [5][6][7][8]17] due to the existence of a third variable, named u. The results in [5][6][7][8]17] are related to systems where the characteristic analysis is standard.…”

“…Our problem differs from the problems in [5][6][7][8]17] due to the existence of a third variable, named u. The results in [5][6][7][8]17] are related to systems where the characteristic analysis is standard. We mean, such papers refer to standard eigenvalues, det(A − λI) = 0, rather than to generalized ones.…”