可积系统在数值算法中的应用

As some intimate relations between certain numerical algorithms and integrable systems have been revealed in recent years, researchers begin to construct numerical algorithms from some existed integrable systems. Such kinds of algorithms are known as the integrable algorithms, which not only produce good numerical results, but also admit some especial geometric and algebraic properties, namely the so-called "integrability". These studies not only promote the development of the theory of integrable systems, but also provide new ideas and methods for computational mathematics, thus enrich the application of integrable systems. Our research focuses on how to design efficient and stable numerical algorithms from integrable systems. On the one hand, study the non-isospectral Toda equation by using Hirota's bilinear method and integrable discretization technique, and then construct a new convergence acceleration algorithm which is effective on the logarithmically convergent sequences. We will also discuss its application in combinatorial mathematics. On the other hand, study the periodic wave solutions of some integrable nonlinear wave equations. By taking full advantage of integrability, we try to design an effective numerical method for solving the corresponding nonlinear wave equations, and then verify the existence of N-periodic wave solutions of these integrable nonlinear wave equations. Besides that, since the current convergence acceleration algorithms designed from integrable systems are applicable for scalar sequences, our research will be devoted to constructing some acceleration algorithms for vector sequences by using the coupled integrable systems.

近年来，学者们注意到某些著名算法与可积系统之间的联系，并且开始从可积系统出发构造新的数值算法。这类算法不仅有良好的数值效果，还具备一些特殊的代数、几何性质，即所谓的“可积性”。这些研究不仅促进了可积系统理论的发展，还为计算数学提供了新的思路和方法，进而丰富了可积系统的应用。本项目将基于可积系统，探讨如何利用系统自身的可积性去构造高效稳定的数值算法。一方面，利用 Hirota 双线性方法和可积离散化技巧研究非等谱的Toda方程，从它的全离散形式出发构造对对数收敛序列有效的收敛加速算法，并探讨非等谱 Toda方程在组合数学中的应用。另一方面研究一些可积的非线性波动方程的多周期波解，充分利用这些方程自身的可积性质设计出有效的数值求解方法。此外，据我们所知目前从可积系统出发构造的收敛加速算法都只适用于数量序列，本项目我们将尝试利用多变量的耦合可积系统设计针对向量序列的收敛加速算法。

本项目主要探讨如何利用可积系统自身的可积性设计高效稳定的数值算法，围绕三个方面展开研究。一，孤子方程N-周期波的数值求解方面：在Nakamura教授提出的直接方法的基础上，我们给出了一种针对KdV型方程三周期波的数值求解算法，并从数值上验证了KdV型方程三周期波解的存在性。二，收敛加速算法的设计方面：基于经典的收敛加速算法epsilon-算法和rho-算法，我们构造了一个新的收敛加速算法，并进行了收敛稳定性分析及数值试验。理论分析和数值结果表明新算法对线性收敛序列和对数收敛序列都是有效的。三，孤子方程的可积离散化及数值应用方面：我们给出了mKdV方程的可积半离散化，然后将时间方向的可积离散与空间方向的拟谱法离散结合在一起，成功构造出一种快速有效求解mKdV方程的数值格式。我们的研究丰富了可积系统在数值算法中的应用，进一步促进了数值算法与可积系统的交叉研究。