High Order Schemes for Gradient Flow with Respect to a Metric

TL;DR

This paper introduces new energy stability criteria for high-order numerical schemes applied to gradient flow equations with respect to a metric, demonstrating their effectiveness on PDEs related to the 2-Wasserstein metric.

Contribution

It develops novel energy stability criteria for multi-step and multi-stage schemes in gradient flow problems, enabling the construction of higher-order energy stable methods.

Findings

Second and third order energy stable schemes are constructed.

The schemes are demonstrated on PDEs involving the 2-Wasserstein metric.

The criteria ensure energy stability for complex evolution equations.

Abstract

New criteria for energy stability of multi-step, multi-stage, and mixed schemes are introduced in the context of evolution equations that arise as gradient flow with respect to a metric. These criteria are used to exhibit second and third order consistent, energy stable schemes, which are then demonstrated on several partial differential equations that arise as gradient flow with respect to the 2-Wasserstein metric.