Dispersive and dissipative properties of discontinuous Galerkin finite element methods for the second-order wave equation
Dispersive and dissipative properties of discontinuous Galerkin finite element methods for the second-order wave equation
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DOI：
10.1007/s10915-005-9044-x
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发表时间：
2006-06-01
影响因子：
2.5
通讯作者：
Muniz, W.
中科院分区：
文献类型：
--
作者：
Ainsworth, M.;Monk, P.;Muniz, W.
当应用于声波方程的空间离散化时，不连续伽辽金有限元法（DGFEM）比标准连续有限元法具有一定的优势。例如，质量矩阵具有块对角结构，该结构与显式时间步进格式结合使用，给出了一种非常经济的时域仿真方案。这个特征是普遍存在的，并延伸到其他时变波动问题，如麦克斯韦方程。计算波传播的一个重要考虑因素是离散化方案与原始系统的色散和耗散特性的比较。我们研究了两种流行的DGFEM格式的这些性质：应用于二阶波动方程的内罚不连续Galerkin有限元法和应用于相应一阶系统的更一般的一类格式。我们展示了如何将多维情况的分析简化为对一维问题的考虑。我们推导了各种格式的色散误差，并推测了其在空间上的推广到高阶近似
Discontinuous Galerkin finite element methods (DGFEM) offer certain advantages over standard continuous finite element methods when applied to the spatial discretisation of the acoustic wave equation. For instance, the mass matrix has a block diagonal structure which, used in conjunction with an explicit time stepping scheme, gives an extremely economical scheme for time domain simulation. This feature is ubiquitous and extends to other time-dependent wave problems such as Maxwell's equations. An important consideration in computational wave propagation is the dispersive and dissipative properties of the discretisation scheme in comparison with those of the original system. We investigate these properties for two popular DGFEM schemes: the interior penalty discontinuous Galerkin finite element method applied to the second-order wave equation and a more general family of schemes applied to the corresponding first order system. We show how the analysis of the multi-dimensional case may be reduced to consideration of one-dimensional problems. We derive the dispersion error for various schemes and conjecture on the generalisation to higher order approximation in space