On the Method of Characteristics in Rate‐Type Viscoelasticity with Non‐Monotone Equilibrium Curve

Abstract

Abstract The standard method of characteristics is applied to a hyperbolic system of partial differential equations describing the motion of a viscoelastic body. The equilibrium curve can be of the van der Waals type such that the system may describe phase transition phenomena. For isolated body problems, sufficient conditions are given on the time integration steps of the first and second numerical approximations such that the total energy of the numerical solution be a nonincreasing function of the number of time integration steps. This property is required since the total energy of the exact solution is a nonincreasing function of time as required by the second law of thermodynamics. Numerical examples of nonisolated body problems when the violations of the above conditions lead to numerical instabilities are presented elsewhere (see Introduction).