Supplementary variable method for structure-preserving approximations to partial differential equations with deduced equations

Abstract

We present a supplementary variable method (SVM) for developing structure-preserving numerical approximations to a partial differential equation system with deduced equations. The PDE system with deduced equations constitutes an over-determined, yet consistent and structurally unstable system of equations. We augment a proper set of supplementary variables to the over-determined system to make it well-determined with a stable structure. We then discretize the modified system to arrive at a structure-preserving numerical approximation to the over-determined PDE system. We illustrate the idea using a dissipative network generating partial differential equation model by developing an energy-dissipation-rate preserving scheme. We then simulate the network generating phenomenon using the numerical scheme. This numerical method is so general that it applies literally to any PDE systems with deduced equations.