Supplementary Variable Method for Developing Structure-Preserving Numerical Approximations to Thermodynamically Consistent Partial Differential Equations

TL;DR

This paper introduces the supplementary variable method, a novel temporal discretization approach that preserves energy dissipation rates in thermodynamically consistent PDE systems, demonstrated on gradient flow models with improved performance.

Contribution

The paper proposes a new supplementary variable method for structure-preserving numerical approximations of thermodynamically consistent PDEs, enhancing energy-dissipation-rate preservation.

Findings

New schemes outperform existing methods in numerical experiments.

The method is applicable to a wide range of thermodynamic models.

Spatial discretization using pseudo-spectral methods complements the temporal scheme.

Abstract

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.