Oscillation waves in Riemann problems for phase transitions
In this work we are interested in one dimensional motion of a body capable of phase transitions. Such a problem is often described by the usual equations of elasticity with non-monotone stress-strain relations [1]. This system of conservation laws is elliptic in the interval of values of the strain where the stress function is monotone decreasing, and is hyperbolic otherwise. This interval is usually referred to as the unstable region. For such systems of mixed type, it has been shown in [2] that a constant state in the elliptic region is unstable in the sense that any local disturbance of a constant state, taken as initial data for the numerical solution, gives rise to a sequence of approximate solutions converging to a non-trivial solution of the Cauchy problem with the constant initial data. This non-trivial solution may be a distributional solution (also called weak solution), or, more generally, a measure-valued solution (abbreviated m-v solution), a concept first introduced by DiPerna [3] (see Definition 2.1). It was also shown that a new type of Riemann waves may appear as (measure-valued) solutions of Riemann problems with data inside the elliptic region. These waves are called oscillation waves, and are associated with persistent oscillations in the numerical solutions of the approximation scheme. To approach the solution of such a 2 × 2 system of conservation laws, we employ a scheme in which each numerical solution approximates the solution of a 3 × 3 system describing the one dimensional motion of a rate-type viscoelastic body (see [4]–[8]) with a Maxwell type viscosity, which is inversely proportional to the grid spacing of the numerical scheme and hence reduces to the original 2 × 2 system as the grid spacing tends to zero. This scheme, based on the method of characteristics proposed in [7], has the mathematical advantage of allowing an energy estimate, which is crucial in the proof of the existence of measure-valued solutions.