Unstable Godunov discrete profiles for steady shock waves

摘要

Given a system of n conservation laws ut + f(u)(x) = 0, the steady shock waves, when processed by the Godunov scheme, admit rather simple discrete profiles. One shows that the linear stability of these profiles depends only on the location of the eigenvalues of some endomorphism of an (n - 1)-dimensional space. Applying our theory to the gas dynamics with the perfect gas law p = (gamma ? 1)rho e, we construct unstable profiles for values of gamma between 1 and gamma* and rather strong shocks. Since gamma* is an element of]7/5, 5/3[, our result applies to the air but not to monoatomic gases. Finally, we illustrate this analysis by some numerical experiments, both with gamma < gamma* and gamma > gamma*. [References: 7]