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

Highlights: • High-order, energy stable, semi-implicit schemes for general gradient flows. • The schemes cost as much as multiple calls per time step to a first-order scheme. • Schemes for gradient flows with varying inner products, such as Wasserstein. • Numerical examples of the new schemes on a variety of gradient flows. 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.