A general strategy for numerical approximations of non-equilibrium models, Part I: thermodynamical systems

Abstract

We present a generic approach to derive numerical approximations for thermodynamical consistent models for nonequilibrium phenomena. This paper focus on thermodynamic models. The central idea behind the systematic numerical approximation is the energy quadratization (EQ) method, where the system energy is quadratized by a new intermediate variable. By applying the EQ strategy, one can develop linear second order in time semi-discrete schemes that preserve the energy dissipation property in the original thermodynamically consistent model equations. The EQ method is developed for time discretization primarily. When coupled with an appropriate spatial discretization, a fully discrete, linear scheme can be developed to warrant the energy dissipation property of the fully discrete scheme. Several examples for multiphase field models are presented to illustrate the usefulness of the approach.