High-order linearly implicit schemes conserving quadratic invariants
High-order linearly implicit schemes conserving quadratic invariants
复制标题
保留二次不变量的高阶线性隐式方案
DOI：
10.1016/j.apnum.2023.02.005
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发表时间：
2023
影响因子：
2.8
通讯作者：
Butcher John C.
中科院分区：
文献类型：
--
作者：
Sato Shun;Miyatake Yuto;Butcher John C.
本文提出了具有二次不变量的常微分方程组的线性隐式和任意高阶守恒型数值格式。许多微分方程都有不变量，保存它们的数值格式已经得到了广泛的研究。由于线性不变量在离散化后可以很容易地保留，所以二次不变量本质上是最简单的。二次不变量是一个重要的对象，不仅出现在许多物理例子中，而且还出现在计算效率较高的一般不变量的守恒格式中，如近年来研究的标量辅助变量方法。众所周知，与一般不变量相比，二次不变量可以相对容易地保持，并且可以用典型的Runge-Kutta方法来保持。然而，对于线性隐式格式和高阶守恒格式的构造，目前还没有统一的方法。在本文中，我们基于典型的Runge-Kutta方法构造了这类格式，并证明了一些与精度有关的性质。
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 maintained relatively easily compared with general invariants, and 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.