Abstract

This paper proposes a framework for constructing high-order linearly implicit exponential integrators that conserve a quadratic invariant. This is then applied to the scalar auxiliary variable (SAV) approach. Quadratic invariants are significant objects that are present in various physical equations and also in computationally efficient conservative schemes for general invariants. For instance, the SAV approach converts the invariant into a quadratic form by introducing scalar auxiliary variables, which have been intensively studied in recent years. In this vein, Sato et al. (Appl. Numer. Math. 187, 71-88 2023) proposed high-order linearly implicit schemes that conserve a quadratic invariant. In this study, it is shown that their method can be effectively merged with the Lawson transformation, a technique commonly utilized in the construction of exponential integrators. It is also demonstrated that combining the constructed exponential integrators and the SAV approach yields schemes that are computationally less expensive. Specifically, the main part of the computational cost is the product of several matrix exponentials and vectors, which are parallelizable. Moreover, we conduct some mathematical analyses on the proposed schemes.

中文翻译：

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保留二次不变量的高阶线性隐式指数积分器及其应用于标量辅助变量方法

摘要

本文提出了一种构建保留二次不变量的高阶线性隐式指数积分器的框架。然后将其应用于标量辅助变量（SAV）方法。二次不变量是存在于各种物理方程以及一般不变量的计算有效的保守方案中的重要对象。例如，SAV方法通过引入标量辅助变量将不变量转换为二次形式，这在近年来得到了深入研究。本着这一精神，佐藤等人。(Appl. Numer. Math. 187 , 71-88 2023) 提出了保留二次不变量的高阶线性隐式方案。在这项研究中，表明他们的方法可以有效地与劳森变换（一种常用于构建指数积分器的技术）合并。还证明了将构建的指数积分器与 SAV 方法相结合可以产生计算成本较低的方案。具体来说，计算成本的主要部分是几个可并行的矩阵指数和向量的乘积。此外，我们对所提出的方案进行了一些数学分析。