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In this paper, we present a novel investigation of the so-called SAVapproach, which is a framework to construct linearly implicit geometricnumerical integrators for partial differential equations with variationalstructure. SAV approach was originally proposed for the gradient flows thathave lower-bounded nonlinear potentials such as the Allen-Cahn andCahn-Hilliard equations, and this assumption on the energy was essential. Inthis paper, we propose a novel approach to address gradient flows withunbounded energy such as the KdV equation by a decomposition of energyfunctionals. Further, we will show that the equation of the SAV approach, whichis a system of equations with scalar auxiliary variables, is expressed asanother gradient system that inherits the variational structure of the originalsystem. This expression allows us to construct novel higher-order integratorsby a certain class of Runge-Kutta methods. We will propose second and fourthorder schemes for conservative systems in our framework and present severalnumerical examples.

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