波动方程的高阶间断有限元方法
High Order Discontinuous Finite Element Method for Wave Equation
DOI: 10.12677/PM.2021.114081, PDF,    科研立项经费支持
作者: 石康康, 马 宁*：中国石油大学(北京)，北京
关键词: 波动方程；间断有限元；高阶多项式；收敛阶；Wave Equation； DG-FEM； Higher Order Polynomial； Convergence Order
摘要: 近年来，间断有限元方法在求解双曲型方程中获得广泛应用，与传统有限元和差分方法相比也有许多优势。在本文中，对双曲型的波动方程应用间断有限元分析，通过构造间断有限元的强形式，选择合适的数值通量，与精确解相比得到了较小的误差以及较高的收敛阶，达到了间断有限元的理论收敛阶。
Abstract: In recent years, discontinuous finite element method has been widely used in solving hyperbolic equations. Compared with traditional finite element method and difference method, discontinuous finite element method has many advantages. In this paper, the discontinuous finite element analysis is applied to the hyperbolic wave equation. By constructing the strong form of the discon-tinuous finite element and selecting the appropriate numerical flux, the smaller error and higher convergence order are obtained compared with the exact solution, and the theoretical convergence order of the discontinuous finite element is reached.
文章引用：石康康, 马宁. 波动方程的高阶间断有限元方法[J]. 理论数学, 2021, 11(4): 669-675. https://doi.org/10.12677/PM.2021.114081

参考文献

[1] Reed, W.H. and Hill, T.R. (1973) Triangular Mesh Methods for the Neutron Transport Equation. Los Alamos Report LA-UR-73-479.
[2] Cockburn, B. and Shu, C.W. (1998) The Runge-Kutta Discontinuous Galerkin Method for Con-servation Laws V: Multidimensional Systems. Journal of Computational Physics, 141, 199-224. [Google Scholar] [CrossRef] 
[3] Cockburn, B. and Shu, C.W. (1989) TVB Runge-Kutta Local Pro-jection Discontinuous Galerkin Finite Element Method for Conservation Laws II: General Framework. Mathematics of Computation, 52, 411-435. [Google Scholar] [CrossRef] 
[4] Hu, F.Q., Hussaini, M.Y. and Rasetarinera, P. (1999) An Analysis of the Discontinuous Galerkin Method for Wave Propagation Problems. Journal of Computational Physics, 151, 921-946. [Google Scholar] [CrossRef] 
[5] Cockburn, B. and Shu, C.W. (2001) Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems. Journal of Scientific Computing, 16, 173-261. [Google Scholar] [CrossRef] 
[6] 章晓. 求解波动方程的近似解析指数时间差分方法及其数值模拟[D]: [硕士学位论文]. 北京: 清华大学, 2013.
[7] 贺茜君. 求解波动方程的间断有限元方法及其波场模拟[D]: [博士学位论文]. 北京: 清华大学, 2015.
[8] 陈二云. 间断有限元数值方法研究及复杂冲击流场的数值模拟[D]: [博士学位论文]. 南京: 南京理工大学, 2008.
[9] 陆金甫, 关治. 偏微分方程数值解法[M]. 北京: 清华大学出版社, 2004.
[10] Shu, C.W. (1988) Total-Variation-Diminishing Time Discretizations. SIAM Journal on Scientific and Statistical Computing, 9, 1073-1084. [Google Scholar] [CrossRef] 
[11] 栾睿琦. 间断有限元地震波场正演模拟[D]: [硕士学位论文] . 北京: 中国石油大学, 2019.
[12] 张铁. 间断有限元理论与方法[M]. 北京: 科学出版社, 2012.

为你推荐