波动方程高精度保结构算法及理论分析

本项目以物理领域中广泛应用的波动方程为研究目标，研究该类方程的高精度算法，建立基于port-Hamiltonian系统的外微分有限元算法的理论框架，主要的设计思想是在间断有限元的离散过程中将每个局部单元处理为port-Hamiltonian系统，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经典波动方程、薛定谔方程、Maxwell方程、Sine-Gordon方程、弹性动力学（elastodynamics）方程等，并将算法应用到诸多实际应用领域，如地震波成像、水波问题、电磁学问题等的数值模拟。这不仅对外微分有限元和port-Hamiltonian系统的发展具有推动作用，而且由于波动方程在众多物理问题中的广泛应用，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 high order discontinuous Galerkin methods for wave equations in physics. We will use exterior calculus which provides a mathematical framework to derive structure preserving numerical discretizations. The key idea is to consider each element in discontinuous Galerkin discretization as a port-Hamiltonian system. 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 classical wave equation, Schrodinger equation, KdV equation, Maxwell equation, Sine-Gordon equation, elastodynamics equations and other related physical models. We will use the new numerical methods to simulate the physical models for seismic imaging, water wave, electromagnetics. The project will be useful for development of the finite element exterior calculus and port-Hamiltonian system. There are important theoretical and practical significance. It will have a good impact and role on many branches of mathematics and interdisciplinary development.