本项目以数学物理中出现的非线性波动问题为研究对象，针对几类具有典型物理意义的方程，如Camassa-Holm方程、 Cahn-Hilliard方程、Hunter-Saxton方程等，设计相应的间断有限元算法，并给出理论分析。.如何处理好该类方程中的强非线性项和非线性色散项，并消除某些没有足够光滑性的解所产生高频色散误差，以及解决高阶导数的时间步长问题，是困扰该类方程数值模拟的主要问题。我们针对这些问题，建立了相应的间断有限元方法以及相关理论。同时，我们研制相关的实用数值模拟软件，用实际算例和理论分析证明了间断有限元方法在求解非线性波动方程过程中的优越性，体现其非线性稳定性、高精度、易于实现并行化、自适应和处理复杂边界等优点。.数值模拟的成功有助于进一步了解非线性演化方程的特性，从而揭示其物理本质，指导数学物理问题研究。

In this project, we consider the discontinuous Galerkin method for the nonlinear wave equations in mathematical physics. These nonlinear wave equations are typical model equations in physics, including Camassa-Holm equation, Cahn-Hilliard equations and Hunter-Saxton equation, etc..The treatment for the strong nonlinear terms and nonlinear dispersive terms, the lack of smoothness at the edge of solutions which introduces high-frequency dispersive errors into the calculation and small time step restriction for stability are the main issues for solving this kind of this problem. It is still a challenge to design stable and accurate numerical schemes. We develop and analysis the discontinuous Galerkin methods. We give the error analysis and perform extensive numerical experiments for nonlinear problems to demonstrate the accuracy and capability of the discontinuous Galerkin methods. These methods are flexible for general geometry, unstructured meshes and hp adaptivity, and have excellent parallel efficiency. They provide a useful class of numerical tools for solving nonlinear wave equations. .The characteristic of the nonlinear evolution equations is further comprehended through the success of the numerical simulation. The essence of the physical phenomena is explored and the research of the mathematical physics will be further developed.

本项目以数学物理中出现的非线性波动问题为研究对象，针对几类具有典型物理意义的方程，如Camassa-Holm方程、 Cahn-Hilliard方程、Hunter-Saxton方程等，设计相应的间断有限元算法，并给出理论分析。如何处理好该类方程中的强非线性项和非线性色散项，并消除某些没有足够光滑性的解所产生高频色散误差，以及解决高阶导数的时间步长问题，是困扰该类方程数值模拟的主要问题。我们针对这些问题，建立了相应的间断有限元方法以及相关理论。同时，我们研制相关的实用数值模拟软件，用实际算例和理论分析证明了间断有限元方法在求解非线性波动方程过程中的优越性，体现其非线性稳定性、高精度、易于实现并行化、自适应和处理复杂边界等优点。数值模拟的成功有助于进一步了解非线性演化方程的特性，从而揭示其物理本质，指导数学物理问题研究。