A general strategy for numerical approximations of non-equilibrium models–Part I: Thermodynamical systems

We present a general approach to deriving energy stable numerical approximations for thermodynamical consistent models for nonequilibrium phenomena. The central idea behind the systematic numerical approximation is the energy quadratization (EQ) strategy, where the sys-tem’s free energy is transformed into a quadratic form by introducing new intermediate variables. By applying the EQ strategy, one can develop linear, high order semi-discrete schemes in time that preserve the energy dissipation property of 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, high order, linear scheme can be developed to warrant the energy dissipation property of the fully discrete scheme. A host of examples for phase field models are presented to illustrate the effectiveness of the general strategy.