非经典双曲型守恒律方程数值模拟与分析

本项目以固体力学和流体力学等领域中出现的重要非经典守恒律方程为研究目标，研究该类方程的高精度间断有限元方法，建立相关理论和严格的数学基础，研制相关的数值模拟软件，为该类方程广泛应用于实际问题提供强有力的工具。研究的方程主要包括非线性双曲-椭圆耦合系统的相变问题、非线性弹性动力学方程及其相关物理应用问题等，并将算法应用到模拟诸多实际应用领域，如复合材料的设计、固体和液体的相变问题、磁力体问题、液体薄膜、半导体问题等。这不仅对非经典守恒律方程的发展具有推动作用，而且由于非经典守恒律方程在数学物理中的典型代表性，将对许多数学分支及交叉学科的发展都有重要影响和促进作用，具有十分重要的理论和现实意义。

In this project, we will develop the high order discontinuous Galerkin methods for the important non-classical hyperbolic conservation law equations in solid and fluids mechanics. We will build up the mathematical theory of the numerical methods and develop the software for the numerical simulations, which will provide the useful tool for solving these type equations in application area. We will mainly focus on the nonlinear hyperbolic-elliptic phase transition equations, nonlinear elastodynamics equations and other related physical models. We will use the new numerical methods to simulate the physical models for complex material, phase transition in solid and fluids mechanics, magneto hydrodynamics, thin liquid films, semi-conductors. The project will be useful for development of the non-classical hyperbolic conservation law equations. There are important theoretical and practical significance. It will have a good impact and role on many branches of mathematics and interdisciplinary development.