Glimm’s Scheme and Conservation Laws of Mixed Type

Abstract

The focus of this paper is on the numerical solution of problems modeled by conservation laws of mixed type. The numerical approximation of a conservation law from nonlinear elasticity (the equation of the motion of an elastic string in the plane), which changes type from hyperbolic to mixed, is considered. An exact solution that will be the test problem is exhibited. Two versions of Glimm’s scheme are investigated. According to the results of Pego and Serre, the first version of Glimm’s scheme does not allow the computation of the solution when it leaves the hyperbolic domain to enter the mixed one because the computed solution is not kinematically admissible. The second version of Glimm’s scheme (the original one) allows the reconstruction of a numerical solution which is kinematically admissible and dynamically admissible on an average but not locally.