论文标题
关于能源定律和runge的稳定性 - 线性半度问题的kutta方法
On Energy Laws and Stability of Runge--Kutta Methods for Linear Seminegative Problems
论文作者
论文摘要
本文提出了一个系统的理论框架，以得出线性半定位系统的一般隐式和显式runge（RK）方法的能量身份。它概括了[Z.中显式RK方法的稳定性分析。太阳和C.-W。 Shu，Siam J. Numer。肛门，57（2019），第1158-1182页。既定的能量身份提供了关于能量在RK离散化中是否以及如何消散的精确表征，从而导致RK方法的弱稳定性标准。此外，我们基于一类对称矩阵的分析性cholesky型分解，发现了所有对角线胶近似的统一能量身份。矩阵的结构非常复杂，从而发现了统一能量身份的发现以及分解的证明高度挑战。我们的证明涉及从高几幅系列理论中的技术组合身份和新技术的构建。我们的框架是由零件技术的整合类似物和连续能量法的一系列扩展所激发的。在某些特殊情况下，我们的分析在连续和离散的能量定律之间建立了密切的联系，从而增强了我们对它们内在机制的理解。给出了隐式方法的几个特定示例，以说明离散的能量定律。一些数值示例进一步证实了理论特性。
This paper presents a systematic theoretical framework to derive the energy identities of general implicit and explicit Runge--Kutta (RK) methods for linear seminegative systems. It generalizes the stability analysis of explicit RK methods in [Z. Sun and C.-W. Shu, SIAM J. Numer. Anal., 57 (2019), pp. 1158-1182]. The established energy identities provide a precise characterization on whether and how the energy dissipates in the RK discretization, thereby leading to weak and strong stability criteria of RK methods. Furthermore, we discover a unified energy identity for all the diagonal Pade approximations, based on an analytical Cholesky type decomposition of a class of symmetric matrices. The structure of the matrices is very complicated, rendering the discovery of the unified energy identity and the proof of the decomposition highly challenging. Our proofs involve the construction of technical combinatorial identities and novel techniques from the theory of hypergeometric series. Our framework is motivated by a discrete analogue of integration by parts technique and a series expansion of the continuous energy law. In some special cases, our analyses establish a close connection between the continuous and discrete energy laws, enhancing our understanding of their intrinsic mechanisms. Several specific examples of implicit methods are given to illustrate the discrete energy laws. A few numerical examples further confirm the theoretical properties.
