波动方程和电磁流体动力学方程的数值解研究

Numerical solutions for three kinds of partial differential equations, including water wave equations, scattering wave equations, and the coupling equations of fluid dynamics and electromagenetics, are of great significance for their practical applications in sciences and engineering. Initial-boundary value problems related to these equations usually experience such restrictions as the unavailability for direct computation, free boundaries and unbounded computational domains. These restrictions lead to great difficulties for the direct application of fundamental numerical methods. This project first aims at the design and establishment of corresponding numerical models and newly-developed algorithms for these equations. These numerical models and algorithms are based on the discontinuous and continuous Galerkin finite element methods and boundary integral equation methods. We will carry out essential theoretical analysis for some methods and models, and perform comprehensive numerical tests to demonstrate their validation.

水波方程，散射波方程以及电磁学与流体动力学的耦合方程这三种数学物理方程的数值解在科学与工程的实际应用中有着重要的意义。与这些方程相关的初边值问题通常具有无法直接计算，边界的自由性和计算区域的无界性等等诸多限制条件。而这些限制条件给基本数值方法的直接应用带来很大的困难。本项目首先针对这些方程建立和设计相应的数值模型和改进的算法。这些数值模型和改进的算法主要建立在间断和连续迦辽金有限元方法和边界积分方程方法基础之上。我们将对这些算法和模型进行必要的理论分析，并通过广泛的数值实验加以验证。

本项目对水波方程，散射波方程以及电磁学与流体动力学的耦合方程开展了如下的研究工作：声波作用问题的算法及理论分析；非线性弱色散浅水波方程及其欧拉方程的保结构算法研究；流固耦合问题的边界积分方程法；重构空腔形状的内部反散射问题的理论及算法研究；二维弹性波外散射问题的伽辽金边界元方法中的超奇异积分算子的研究等等。这些研究成果完善和丰富了现有理论和数值方法，对自然灾害的防治，通讯成像医疗等实际应用问题有一定指导意义。本项目总共发表了15篇Sci论文，1篇会议论文.培养已毕业博士5人，硕士3人。