Scalar auxiliary variable approach for conservative/dissipative partial differential equations with unbounded energy

TL;DR

This paper extends the scalar auxiliary variable (SAV) approach to handle gradient flows with unbounded energy, enabling the construction of higher-order geometric integrators for a broader class of PDEs.

Contribution

It introduces a novel energy decomposition method within the SAV framework to address unbounded energies and develops higher-order integrators for such systems.

Findings

Successfully applied to KdV equation with unbounded energy

Developed second and fourth order schemes for conservative systems

Numerical examples demonstrate effectiveness and accuracy

Abstract

In this paper, we present a novel investigation of the so-called SAV approach, which is a framework to construct linearly implicit geometric numerical integrators for partial differential equations with variational structure. SAV approach was originally proposed for the gradient flows that have lower-bounded nonlinear potentials such as the Allen-Cahn and Cahn-Hilliard equations, and this assumption on the energy was essential. In this paper, we propose a novel approach to address gradient flows with unbounded energy such as the KdV equation by a decomposition of energy functionals. Further, we will show that the equation of the SAV approach, which is a system of equations with scalar auxiliary variables, is expressed as another gradient system that inherits the variational structure of the original system. This expression allows us to construct novel higher-order integrators by a certain class of Runge-Kutta methods. We will propose second and fourth order schemes for conservative systems in our framework and present several numerical examples.