Optimal Discontinuous Galerkin Methods for the Acoustic Wave Equation in Higher Dimensions
Optimal Discontinuous Galerkin Methods for the Acoustic Wave Equation in Higher Dimensions
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DOI：
10.1137/080729062
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发表时间：
2009-10
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影响因子：
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通讯作者：
Eric T. Chung;B. Engquist
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文献类型：
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作者：
Eric T. Chung;B. Engquist
本文提出并分析了一类新的不连续伽辽金（DG）方法来求解混合形式的声波方程。传统的混合有限元方法产生了节能方案，但这些方案是隐式的，使得时间步进效率低下。标准DG方法给出了显式格式，但这些方法通常是耗散的或次优收敛的，这取决于数值通量的选择。我们的新方法可以被看作是这两种技术之间的折衷，因为它既明确又节约能源，在当地和全球都是如此。此外，它还可以看作是Raviart-Thomas有限元法和有限体积法的推广版本。严格分析了新方法的稳定性和收敛性，并证明了该方法是最优收敛的。此外，为了将新方法应用于无界区域，我们将新方法应用于一阶吸收边界条件。分析了所得数值格式的稳定性。
In this paper, we develop and analyze a new class of discontinuous Galerkin (DG) methods for the acoustic wave equation in mixed form. Traditional mixed finite element (FE) methods produce energy conserving schemes, but these schemes are implicit, making the time-stepping inefficient. Standard DG methods give explicit schemes, but these approaches are typically dissipative or suboptimally convergent, depending on the choice of numerical fluxes. Our new method can be seen as a compromise between these two kinds of techniques, in the way that it is both explicit and energy conserving, locally and globally. Moreover, it can be seen as a generalized version of the Raviart-Thomas FE method and the finite volume method. Stability and convergence of the new method are rigorously analyzed, and we have shown that the method is optimally convergent. Furthermore, in order to apply the new method for unbounded domains, we apply our new method with the first order absorbing boundary condition. The stability of the resulting numerical scheme is analyzed.