High-order supplementary variable methods for thermodynamically consistent partial differential equations

Abstract

In recent years, there has been growing interest in developing numerical methods to retain consistency between the partial differential equation system and its deduced equations (e.g., the thermodynamically consistent model and its energy law) after numerical approximations. A supplementary variable method (SVM) of second-order was recently introduced in [1,2] to preserve thermodynamic consistency for thermodynamically consistent models. In this paper, we present a new class of high-order supplementary variable methods. These methods consist of two parts: firstly, the original model is discretized using a high-order prediction method, and secondly, the SVM reformulated system is approximated by a high-order correction method. The newly developed schemes are rigorously proven to preserve high-order accuracy with increased stability through the use of a stabilizer. Additionally, the schemes inherently maintain thermodynamic consistency for thermodynamically consistent models, retaining the original energy law at the discrete level. To assess the high-order supplementary variable methods, we apply them to two thermodynamically consistent partial differential equation systems: the functionalized Cahn–Hilliard model and the ternary Cahn–Hilliard model. We then provide several benchmark examples to demonstrate the accuracy, stability, and efficiency of the newly developed high-order numerical methods.