The Onsager principle and structure preserving numerical schemes

TL;DR

This paper introduces a framework based on the Onsager principle for designing energy-stable numerical schemes that preserve physical properties in gradient flow systems, unifying and extending existing methods.

Contribution

It establishes a natural framework leveraging the Onsager principle for constructing energy-stable, structure-preserving numerical schemes for gradient flow PDEs.

Findings

Several widely used schemes are derived naturally within this framework

The approach demonstrates versatility across diverse gradient flow systems

The schemes uphold crucial physical properties such as energy stability

Abstract

We present a natural framework for constructing energy-stable time discretization schemes. By leveraging the Onsager principle, we demonstrate its efficacy in formulating partial differential equation models for diverse gradient flow systems. Furthermore, this principle provides a robust basis for developing numerical schemes that uphold crucial physical properties. Within this framework, several widely used schemes emerge naturally, showing its versatility and applicability.