Oscillation waves in Riemann problems inside elliptic regions for conservation laws of mixed type

Riemann problems with initial data inside elliptic regions are quite different from those for hyperbolic systems. First, we have found that approximate solutions may present persistent oscillations, giving rise to a new type of (measure-valued) waves besides the usual (distributional) ones, shocks and rarefaction waves. Second, any local disturbance of a constant state inside the elliptic region will result in a non-trivial (distributional or, more generally, measure-valued) solution, which is independent of any particular choice of disturbance. For our numerical experiments, we establish two analytical results for testing convergence of finite difference schemes, and for determining expectation values of state functions with respect to the measure-valued solutions when oscillation waves occur. Numerical examples are presented to illustrate those interesting aspects, including the appearance of oscillation waves together with the analysis of the corresponding Young measures.

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