The generalized Riemann problem for a simplified model of one-dimensional ideal gas in magnetogasdynamics in a neighborhood of the origin(t>0)in the(x,t)plane is considered. According to the different cases of the corresponding Riemann solutions, we construct the perturbed solutions uniquely with the characteristic method. We find that, for some case, the contact discontinuity appears after perturbation while there is no contact discontinuity of the corresponding Riemann solution. For most cases, the Riemann solutions are stable and the perturbation can not affect the corresponding Riemann solutions. While, for some few cases, the forward (backward) rarefaction wave can be transformed into the forward (backward) shock wave which shows that the Riemann solutions are unstable under such local small perturbations of the Riemann initial data.
The Riemann problem for one-dimensional isentropic flow of a mixture of a non-ideal gas with small solid particles