High order, semi-implicit, energy stable schemes for gradient flows

TL;DR

This paper presents high order, semi-implicit Runge-Kutta schemes for gradient flows, ensuring energy stability and applicability to complex PDEs like Wasserstein gradient flows, advancing numerical methods for evolution equations.

Contribution

Introduces a new class of high order, energy stable, semi-implicit schemes for gradient flows, including those with solution-dependent inner products, extending stability and accuracy.

Findings

Schemes are unconditionally stable and high order accurate.

Applicable to a variety of gradient flows, including Wasserstein-based PDEs.

Demonstrated energy stability across different gradient flow models.

Abstract

We introduce a class of high order accurate, semi-implicit Runge-Kutta schemes in the general setting of evolution equations that arise as gradient flow for a cost function, possibly with respect to an inner product that depends on the solution, and we establish their energy stability. This class includes as a special case high order, unconditionally stable schemes obtained via convexity splitting. The new schemes are demonstrated on a variety of gradient flows, including partial differential equations that are gradient flow with respect to the Wasserstein (mass transport) distance.