Discrete shock profiles for scalar conservation laws with discontinuous fluxes

The paper considers a scalar conservation law with a discontinuous flux F of the form F(x,u)=H(x)g(u)+(1−H(x))h(u) where H(x) is the Heaviside function. Herein, the fluxes g and h are supposed to have one minimum and no maximum and at most one crossing in the interior of the domain of definition. The aim is to verify a weak solution of such a problem in the following way: We are looking for discrete shock profiles for its continuously differentiable perturbation with a parameter ε and Godunov's scheme for conservation laws with spatially varying flux functions. The obtained discrete shock profile satisfies a discrete entropy condition of Kruzkhov type and after letting ε→0, approaches an entropy weak solution of the original equation.