Instabilities in Glimm’s Scheme for Two Systems of Mixed Type

Abstract

The behavior of Glimm’s scheme is examined when applied to two systems from nonlinear elasticity, which change type from hyperbolic to elliptic or mixed type, i.e., a pair of wave speeds coalesce and turn complex. The hyperbolic regime is invariant for Glimm’s scheme, but not for smooth solutions, so strong convergence is not possible in general. Some simple exact solutions for these systems are considered, which illustrate how the approximate solutions are affected by the oscillatory instabilities in these systems. The approximations exhibit macroscopically random transitions to new states and nonconvergence. In one example, however, convergence in expectation is observed, and weak convergence can be observed if the Courant–Friedrichs–Levy (CFL) number is fixed.