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.
A new symmetric linearly implicit exponential integrator preserving polynomial invariants or Lyapunov functions for conservative or dissipative systems
Conservative and dissipative discretisations of multi-conservative ODEs and GENERIC systems
Mathematical Institute, University of Oxford（牛津大学数学研究所）；Mathematical Institute, Faculty of Mathematics and Physics, Charles University（查理大学数学物理学院数学研究所）
A general framework to construct schemes satisfying additional conservation relations. Application to entropy conservative and entropy dissipative schemes
Institute of Mathematics, University of Zurich（苏黎世大学数学研究所）
