Structure preserving approximation of dissipative evolution problems

TL;DR

This paper introduces a unified framework for numerically approximating dissipative evolution problems, ensuring the schemes preserve energy or entropy structures across various applications.

Contribution

It develops a general variational approach that guarantees numerical schemes inherit dissipative properties for a broad class of evolution problems.

Findings

Galerkin methods preserve dissipative behavior

Framework applies to diffusive, Hamiltonian, and gradient systems

Numerical schemes inherit energy or entropy structures

Abstract

We present a general abstract framework for the systematic numerical approximation of dissipative evolution problems. The approach is based on rewriting the evolution problem in a particular form that complies with an underlying energy or entropy structure. Based on the variational characterization of smooth solutions, we are then able to show that the approximation by Galerkin methods in space and discontinuous Galerkin methods in time automatically leads to numerical schemes that inherit the dissipative behavior of the evolution problem. The proposed framework is rather general and can be applied to a wide range of applications. This is demonstrated by a detailed discussion of a variety examples ranging from diffusive partial differential equations to Hamiltonian and gradient systems.