关于分裂步B样条配置方法的算法实现和理论研究

Numerical methods for nonlinear multidimensional partial differential equations are well known as their complexities and huge computational costs. How to solve these problems quickly and effectively is significant and valuable for researches and applications. This project is devoted to construct efficient split-step B-spline collocation methods by combining the split-step technique with the modified classical B-spline collocation method. The proposed method is implemented and applied to solve the nonlinear multidimensional partial differential equations with high efficiency, such as the Ginzburg-Landau equation, the Schrodinger equation and the problem of Bose-Einstein condensates. The series expansion of operators, the error estimate by norms and the mathematical induction are applied to analyze and research the accuracy, convergence and stability of the split-step B-spline collocation method. The research methodological system is constructed as the theoretical support and method guidance when the present method is applied to the field of numerical methods for partial differential equations, and even to physics and engineering.

非线性多维偏微分方程（组）问题的数值求解，因方程的复杂性和计算规模宏大而著称。如何快速有效的求解该类问题具有重要的研究意义和应用价值。本项目拟将分裂步技术与B样条配置方法有机结合，在改进传统B样条配置方法的基础上，构造切实可行的分裂步B样条配置方法。对该方法进行算法实现，使其能够高效率求解多维非线性偏微分方程（组）问题，如金兹堡-朗道方程、薛定谔方程（组）及玻色-爱因斯坦凝聚问题等。本项目拟采用算子级数展开法、误差范数估计法和数学归纳法等，分析和研究分裂步B样条配置方法的算法精度、收敛性和稳定性等，建立理论研究体系，为该方法在微分方程数值解领域，甚至在物理学、工程学等领域的广泛应用提供理论基础和方法指导。

在物理学、生物学和工程学等相关领域的数学模型通常都是多维非线性偏微分方程。本项目将分裂步技术和B样条配置方法相结合成功构造出分裂步B样条配置方法，该方法将多维问题分解成若干一维问题，从而降低多维问题的计算量。分裂步B样条配置方法被用于求解二维和三维非线性薛定谔方程等，这些方程问题可以包含Dirichlet型边值条件，也可以包含Neumann型边值条件。数值实验验证了分裂步B样条配置方法的守恒性和收敛性等相关数值性能。该类方法也被应用于模拟玻色-爱因斯坦凝聚问题。本项目为多维非线性问题的研究提供了一类有效的数值方法。