基于广义交替数值通量的局部间断Galerkin方法求解二维波动方程  

Two-dimensional wave equation solved by generalized alternating flux based local discontinuous Galerkin method

作　　者：张荣培 王迪 蔚喜军[2] 温学兵[1] Zhang Rong-Pei;Wang Di;Yu Xi-Jun;Wen Xue-Bing(College of Mathematics and Systems Science,Shenyang Normal University,Shenyang 110034,China;Institute of Applied Physics and Computational Mathematics,Beijing 100088,China)

机构地区：[1]沈阳师范大学数学与系统科学学院,沈阳110034 [2]北京应用物理与计算数学研究所,北京100088

出　　处：《物理学报》2020年第2期54-60,共7页Acta Physica Sinica

基　　金：国家自然科学基金(批准号:11571002);国防科技重点实验室基金(批准号:6142A0502020717);辽宁省自然科学基金(批准号:201805509960);中国工程物理研究院科学基金(批准号:2013A0202011,2015B0101021)资助的课题~~

摘　　要：波的传播往往在复杂的地质结构中进行,如何有效地求解非均匀介质中的波动方程一直是研究的热点.本文将局部间断Galekin(local discontinuous Galerkin, LDG)方法引入到数值求解波动方程中.首先引入辅助变量,将二阶波动方程写成一阶偏微分方程组,然后对相应的线性化波动方程和伴随方程构造间断Galerkin格式;为了保证离散格式满足能量守恒,在单元边界上选取广义交替数值通量,理论证明该方法满足能量守恒性.在时间离散上,采用指数积分因子方法,为了提高计算效率,应用Krylov子空间方法近似指数矩阵与向量的乘积.数值实验中给出了带有精确解的算例,验证了LDG方法的数值精度和能量守恒性;此外,也考虑了非均匀介质和复杂计算区域的计算,结果表明LDG方法适合模拟具有复杂结构和多尺度结构介质中的传播.The wave propagation is often carried out in complex geological structures. Solving the wave propagation problem effectively in inhomogeneous medium is of great interest and has many applications in physics and engineering. In this paper, the local discontinuous Galekin(LDG) method is applied to the numerical solution of the second-order wave equation. Firstly, the auxiliary variables are introduced, and the second-order wave equations are written as a system of first-order partial differential equations. Then the discontinuous Galerkin format is applied to the corresponding linearized wave equations and adjoint equations. We consider the triangulation in this paper. In order to ensure that the discrete format satisfies the energy conservation, the generalized alternating flux is chosen on the element boundary. We proves that the LDG method satisfies the energy conservation. The exponential integral factor method is used in time discretization. In order to improve the computational efficiency, the Krylov subspace method is used to approximate the product of the exponential matrix and the vector. Numerical examples with exact solutions are given in numerical experiments. The numerical results verify the numerical precision and energy conservation of the LDG method. In addition, the calculation of inhomogeneous medium and complex computational regions are considered. The results show that the LDG method is suitable for simulation of complex structures and propagation in multi-scale structured medium.

关 键 词：波动方程  广义交替数值通量  局部间断Galerkin方法  能量守恒 

分 类 号：O241.8[理学—计算数学]