  We present a new temporal discretization paradigm for developing
energy-production-rate preserving numerical approximations to thermodynamically
consistent partial differential equation systems, called the supplementary
variable method. The central idea behind it is to introduce a supplementary
variable to the thermodynamically consistent model to make the over-determined
equation system, consisting of the thermodynamically consistent PDE system, the
energy definition and the energy dissipation equation, structurally stable. The
supplementary variable allows one to retain the consistency between the energy
dissipation equation and the PDE system after the temporal discretization. We
illustrate the method using a dissipative gradient flow model. Among virtually
infinite many possibilities, we present two ways to add the supplementary
variable in the gradient flow model to develop energy-dissipation-rate
preserving algorithms. Spatial discretizations are carried out using the
pseudo-spectral method. We then compare the two new schemes with the energy
stable SAV scheme and the fully implicit Crank-Nicolson scheme. The results
favor the new schemes in the overall performance. This new numerical paradigm
can be applied to any thermodynamically consistent models.
