Obtaining shock solutions via Maslov's theory and Colombeau's algebra for conservation laws with analytical coefficients

S. Bernard, A. Méril, P. Rodr\'\i guez-Bermúdez, B. Valiño-Alonso

Via the algebra of generalized functions, we investigate the generalized Riemann's problem associated to conservation laws with analytical coefficients. This allows us to transform the problem into a system of ordinary differential equations. In some particular cases, such as Burgers' and conservative Richard's equation, approximated solutions are obtained by the truncation of the so-called Hugoniot--Maslov's chain and numerical simulations are also presented in the case of equations with polynomial coefficients.