论文标题
高阶线性隐式方案保存二次不变
High-order linearly implicit schemes conserving quadratic invariants
论文作者
论文摘要
在本文中，我们建议使用二次不变的普通微分方程线性隐式和任意的高阶保守数值方案。许多微分方程都具有不变的，并且已广泛研究了用于保存它们的数值方案。由于在离散化后可以轻松地保持线性不变性，因此二次不变性本质上是最简单的。二次不变性是重要的对象，不仅在许多物理示例中，而且在计算有效的保守计划中，近年来已经研究了诸如标量辅助变量方法的一般不变性方法。众所周知，与一般不变性相比，二次不变性可以保持相对容易，并且确实可以通过规范runge-kutta方法保留。但是，没有统一的方法来构建线性隐式和高阶保守方案。在本文中，我们基于规范runge-Kutta方法构建了此类方案，并证明了一些涉及准确性的属性。
In this paper, we propose linearly implicit and arbitrary high-order conservative numerical schemes for ordinary differential equations with a quadratic invariant. Many differential equations have invariants, and numerical schemes for preserving them have been extensively studied. Since linear invariants can be easily kept after discretisation, quadratic invariants are essentially the simplest ones. Quadratic invariants are important objects that appear not only in many physical examples but also in the computationally efficient conservative schemes for general invariants such as scalar auxiliary variable approach, which have been studied in recent years. It is known that quadratic invariants can be kept relatively easily compared to general invariants, and indeed can be preserved by canonical Runge--Kutta methods. However, there is no unified method for constructing linearly implicit and high order conservative schemes. In this paper, we construct such schemes based on canonical Runge--Kutta methods and prove some properties involving accuracy.
