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.
翻译:在本文中,我们提出了对所谓的SAV方法的新调查,这是为具有变异结构的局部差异方程式建立线性隐含的几何数字集成器的框架。SAV方法最初是为具有诸如Allen-Cahn和Cahn-Hilliard等较低范围非线性潜力的梯度流提出的,关于能源的这一假设至关重要。在本文中,我们提出了一种新颖的方法,以解决具有无约束能量的梯度流,例如通过分解能源功能的KdV等式。此外,我们将表明SAV方法的等式是带有卡路里辅助变量的方程式,它作为另一个继承原系统变异结构的梯度系统,以另一种梯度系统的形式表达出来。这使我们能够用某种Runge-Kutta方法来建造新的更高顺序的梯度集成器。我们将在我们的框架中提出保守系统的第二和第三顺序计划,并提出几个数字例子。