移动区域内守恒律方程的高精度数值模拟方法与分析

In this project, we will develop the discontinuous Galerkin (DG) methods for the conservation laws with deformable domains, moving boundaries or free surface, in the framework of the Arbitrary Lagrangian-Eulerian (ALE) description, including the construction of the basis of spatial discretization, the time integration methods, the shock and bound limiters, and the mesh moving velocity field. The deformable domains are taken into account in the ALE description, which gives the opportunity to track the free surface and material interface adaptively and preserving high order accuracy. Another motivation of using the ALE description is the Galiliean invariant for some physical problems，for example cosmological hydrodynamical simulations. Numerically this property could be preserved under the moving mesh. To achieve the high-order accuracy, the basic requirement is that any ALE computational method should be able to predict exactly the trivial solution of a uniform flow, i.e. geometric conservation law. The important requirement of the numerical method in our project is that it is able to preserve the geometric conservation law also. Meanwhile, the conservation laws admit discontinuous solutions (shocks), which will cause the spurious oscillation numerically. We need to develop the shock limiters in moving grids to suppress the oscillation and maintain high order accuracy. Bound limiter is also necessary for some physical problems, for example the mass and energy are always positive in the Euler equations. The project will be useful in high-order accuracy simulations of physical problems with deformable domain, moving boundary，free surface and fluid-structure problems.

本项目主要目标是构造移动区域内守恒律方程在任意拉格朗日－欧拉（ALE）坐标下的高精度间断有限元数值模拟方法，包括有限元空间的构造、时间离散方法、振荡和有界限制器，以及网格速度场的生成。ALE坐标的优势在于网格能够高精度追踪自由界面和移动界面，并达到自适应的效果。某些物理问题具有伽里略不变性，ALE坐 标下网格随参考坐标移动使得数值上保持伽里略不变性成为可能。由于数值模拟时的精度会也受到网格速度场的影响，本项目的主要任务就是构造精度不受网格 速度场影响的高精度间断有限元方法，即满足"几何守恒律方程"。由于守恒律方程允许间断解的存在，为避免数值振荡，本项目的另一主要任务为，构造振荡限制器，以保证数值格式的稳定性。为保证物理问题质量、能量等非负的性质，构造ALE坐标下的有界限制器，也是数值方法实际应用的重要保证。本项目将会移动区域、移动边界、自由面和流固耦合等问题的高精度数值模拟起到推动作用。

本项目针对移动区域下的守恒律方程，构造并分析了任意Lagrangian-Eulerian(ALE)坐标下的间断有限元(DG)方法。主要内容包括ALE坐标下的几何守恒律，这是保证数值格式能够达到高精度的基础，也是应用各种限制器的前提条件；数值格式的稳定性、收敛性与误差分析；守恒律相关问题的应用，包括Euler方程、Hamilton-Jacobi方程等。本项目发表标注论文13篇 ，其中SCI收录11篇，主要代表作发表在Mathematics of Computation, Journal of Scientific Computing, Journal of Computational Physics, Journal of Computational and Applied Mathematics, Communications in Computational Physics等专业顶级杂志。培养博士 生2名，硕士生1名。