Non-monotonic traveling wave and computational solutions for gas dynamics Euler equations with stiff relaxation source terms

We study the existence of non-monotone traveling wave solutions and its properties for an isothermal Euler system with relaxation describing the perfect gas flow. In order to confront our results, we first apply a mollification approach as an effective regularization method for solving an ill-posed problem for an associated reduced system for the Euler model under consideration, which in turn is solved by using the method of characteristics. Next, we developed a cheap unsplitting finite volume scheme that reproduces the same traveling wave asymptotic structure as that of the Euler solutions of the continuous system at the discrete level. The method is conservative by construction and relatively easy to understand and implement. Although we do not have a mathematical proof that our designed scheme enjoys the asymptotic preserving and well-balanced properties, we were able to reproduce consistent solutions for the more general Euler equations with gravity and friction recently published in the specialized literature, which in turn are procedures based on a Godunov-type scheme and based on an asymptotic preserving scheme, yielding good verification and performance to our method.