拟插值格式构造及其在非线性偏微分方程数值计算中的应用研究

The nonlinear partial differential equation is an important bridge between many branches of pure mathematics and the nonlinear problems in the fields of natural science and engineering, especially the nonlinear dispersive wave equation with the peak soliton solution, is of great significance in both physical and practical applications. The quasi-interpolation scheme has attracted more and more attention due to its relatively high accuracy, stable computation, flexible node configuration, small computational effort and easy computation, especially without the need to solve linear equations, thus its application is expanding. Therefore, it is of great theoretical and practical value to solve the nonlinear partial differential equation by using the quasi-interpolation scheme..Our research is mainly focused on the construction and improvement of the quasi-interpolation scheme, and its application in the numerical computation of the nonlinear partial differential equation, especially the nonlinear dispersive wave equation with the peak soliton solution. The main contents include the construction of the quasi-interpolation scheme, the improvement of the quasi-interpolation scheme and the application of the quasi-interpolation scheme to the numerical solution of the nonlinear partial differential equation. In other words, the quasi-interpolation scheme is used to approximate the spatial derivatives in the equation, the time discrete method is combined to solve the partial differential equation and the corresponding numerical algorithm is presented. Finally, the theoretical system of solving the nonlinear partial differential equation based on the quasi-interpolation scheme is further established.

非线性偏微分方程是纯粹数学的许多分支和自然科学及工程技术等领域中非线性问题之间的一座重要的桥梁，尤其是具有尖峰孤立子解的非线性色散波方程，在物理和实际应用中都有重要的意义。拟插值格式因其精度相对较高、计算稳定、节点配置灵活、计算工作量小、计算格式简单等优点，尤其是不需要解线性方程组的特点，越来越引起人们的关注，应用不断拓展。因此，利用拟插值格式来求解非线性偏微分方程具有非常重要的理论意义和实用价值。.我们的研究主要围绕拟插值格式的构造、改进及其在非线性偏微分方程（特别是具有尖峰孤立子解的非线性色散波方程）数值计算中的应用展开，主要内容包括拟插值格式的构造、拟插值格式的改进、拟插值格式在非线性偏微分方程数值解中的应用，也就是说用拟插值格式近似方程中的空间导数项，结合时间离散方法，从而数值求解偏微分方程，并给出相应的数值算法，进一步建立基于拟插值格式数值求解非线性偏微分方程的理论体系。

在物理学、流体力学、生态学和生物学等诸多领域内提出了大量的非线性数学问题，这些问题在数学理论上往往是通过具有奇异和退化的一些非线性抛物方程(组)和非线性波方程(组)等非线性发展方程来描述。本项目主要围绕非线性偏微分方程及其数值求解展开。在本项目的资助下取得的研究成果有（1）给出了一类四阶非线性偏微分方程的有限差分法。（2）研究了一类非线性高阶退化拟抛物方程的有限差分方法。（3）证明了四阶双退化抛物方程解的存在性和唯一性。（4）研究了一类非线性四阶偏微分方程的精确解。（5）在国内外核心刊物与国际会议上发表高质量学术论文5篇，其中1篇SCI检索，1篇EI检索。（6）进一步加强了与同行的合作交流，参加了一次偏微分方程会议。（7）指导了6名硕士研究生，其中4位在读，2位已经顺利毕业。