耗散型非局部发展方程的高效数值算法及其长时间性态

分数阶复Ginzburg-Landau方程是一类带非局部特征的非线性耗散型发展方程，在科学与工程中广受关注。当前针对该方程的数值分析研究主要侧重于算法在有限时间区间的计算效率和收敛性分析。然而，该方程为无穷维耗散系统，具有长时间动力学性态。本项目旨在研究可用于长时间计算的数值算法，从而在数值仿真中更好地捕捉方程的长时间性态。为此，我们将构造几类保持原系统耗散性的数值算法，特别是保耗散的高阶算法。同时，为有效评估该类算法的长时间性态，我们将考察算法的长时间稳定性和收敛性。另外，在保耗散算法的构造中，我们将给出空间半离散系统保持原系统耗散性结构的充分或充要条件，为分数阶Laplacian算子构造适当的离散方法。本项目所获研究成果将被广泛应用于极化子凝聚、相流传输以及神经网络等科学问题，并为应用中所需的长时间数值仿真提供必要的技术支持和理论依据。

The fractional complex Ginzburg-Landau equation is a class of nonlinear dissipative evolution equations with the nonlocal character, and it has received lots of attention in scientific and engineering fields. So far, the existing numerical studies mainly focus on the efficiency and convergence of numerical methods in the finite time interval. However, the equation is an infinite-dimensional dissipative system and has long-time dynamics. In order to capture the long-time dynamics in numerical simulations, in this project, we will investigate in detail the long-time behaviour of numerical methods. To this end, we will develop some dissipativity-preserving numerical methods, especially the high-order methods. Meanwhile, for evaluating the behaviour of these methods, we will study their long-time stability and convergence. Furthermore, we will give a necessary and sufficient condition to ensure that the semi-discretized system preserves the dissipativity of the original system. Based on this condition, we will develop some new numerical approximations to the fractional Laplacian. The research results in this project will provide technical support and theoretical basis for the long-time simulation of the equation in polariton condensates, phase transitions, neural networks and other fields of science.