三类非线性偏微分方程的高精度差分方法及其理论研究

Nonlinear partial differential equations are important mathematical model which have been applied in many fields, such as viscoelasticity, nuclear reaction dynamics, mechanics, fluid mechanics and so on. The high-order accuracy numerical algorithm for solving nonlinear partial differential equations can reduce the computational cost and improve the computational efficiency. It has important theoretical significance and application value. This project mainly considers the numerical solutions of the following problems: Burgers equation,Dirac equation and Stefan problem. The techniques of combining Cole-Holf transformation, time-splitting, interface-tracing and boundary immobilization method with the finite difference method will be applied to construct the high-order accuracy numerical algorithms for the three kinds of the nonlinear partial differential equations, respectively; For the multi-dimensional problems, the high-order accuracy ADI schemes will be constructed to reduce the computational cost and improve the computational efficiency; The conservation, unique solvability, stability and convergence of the algorithms will be analyzed rigorously by discrete energy method; Moreover, numerical experments will be used to solve problems in application, to verify the theory of the numerical schemes and to optimize algorithms. To present the high-order accuracy numerical schemes and develop their theory for three kinds of nonlinear partial differential equations is the ultimate goal of this project.

非线性偏微分方程是粘弹性力学、核反应动力学、生物力学、流体力学等诸多领域中具有重要应用价值的数学模型。发展求解非线性偏微分方程的高精度数值方法可以降低计算成本、提高计算效率，是具有重要理论意义和应用价值。本项目拟考虑Burgers方程、Dirac方程、Stefan问题等三类非线性偏微分方程的数值求解。尝试运用Cole-Holf变换、 时间分裂法、边界追踪法或边界固定法等与有限差分法相结合的方法分别对三类方程建立高精度数值方法；对高维问题构造高精度的ADI格式来降低计算的存储量和复杂性，提高计算效率；利用离散能量法对所建立的数值格式的守恒性以及解的存在唯一性、稳定性和收敛性给出严格的理论分析；通过数值求解应用领域中的相关问题验证数值格式的理论结果, 并优化算法。构造求解这三类非线性偏微分方程的高精度数值格式并发展其基本理论是本项目的最终目标。

非线性偏微分方程在粘弹性力学、核反应动力学、生物力学、流体力学等诸多领域中具有重要应用价值的数学模型；分数阶微分方程被广泛地应用于光学与热学系统、流变学及材料和力学系统、信号处理和系统识别、控制和机器人及其他应用领域。 本项目分别对非线性Burgers方程、非线性时间分数阶四阶反应扩散方程、一维及二维多项时间分数阶波方程、时空分数阶偏微分方程以及时间分数阶次扩散方程等建立高精度的数值求解格式。针对分数阶导数的非局部性和全局依赖性导致的巨大的计算量和存储量，在构造高精度差分格式的同时，设计了快速的数值格式。对于高维问题构造高精度的快速的交替方向隐格式(ADI格式)来降低计算的存储量和复杂性，提高计算效率；利用离散能量法对所建立的数值格式的解的存在唯一性、稳定性和收敛性给出严格的理论分析；通过数值求解应用领域中的相关问题验证数值格式的理论结果,并优化算法。