Lie group analysis and Riemann problems for a 2 × 2 system of balance laws

The aim of this paper is to present two different approaches, both based on Lie symmetry analysis, allowing to obtain solutions of the Riemann problem with non-classical discontinuous initial data for a system of balance laws. The first method uses the invariance with respect to a Lie group of exact point symmetries, whereas the second one is based on the approximate Lie symmetries. The two methods are applied to a particular 2 × 2 system of balance laws describing rate-type materials. Moreover, some numerical simulations are performed in order to validate the approximate solutions and check that the exact solution, as a parameter involved in the initial conditions tends to zero, approaches the solution of the classical Riemann problem.