Energy conserving Galerkin approximation of two dimensional wave equations with random coefficients
Energy conserving Galerkin approximation of two dimensional wave equations with random coefficients
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DOI：
10.1016/j.jcp.2018.12.018
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发表时间：
2018-05
期刊：
影响因子：
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通讯作者：
Ching-Shan Chou;Yukun Li;D. Xiu
中科院分区：
文献类型：
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作者：
Ching-Shan Chou;Yukun Li;D. Xiu
非均匀介质中的波传播问题在物理和工程中有着广泛的应用。最近，由于不确定性，这可能会出现从介质的杂质的随机效应，已经有越来越多的兴趣。本文考虑了一个二维随机系数波动方程，该方程在空间上可能是不连续的。将广义多项式混沌方法与随机Galerkin近似相结合，采用局部间断Galerkin方法进行空间离散。我们的方法被证明是能量保持在半离散形式，以及在全离散形式，当使用蛙跳时间离散。它的收敛速度被证明是最佳的，误差线性增长的时间。数值试验验证了该格式的理论性能。
Wave propagation problems for heterogeneous media are known to have many applications in physics and engineering. Recently, there has been an increasing interest in stochastic effects due to the uncertainty, which may arise from impurities of the media. This work considers a two-dimensional wave equation with random coefficients which may be discontinuous in space. Generalized polynomial chaos method is used in conjunction with stochastic Galerkin approximation, and local discontinuous Galerkin method is used for spatial discretization. Our method is shown to be energy preserving in semi-discrete form as well as in fully discrete form, when leap-frog time discretization is used. Its convergence rate is proved to be optimal and the error grows linearly in time. The theoretical properties of the proposed scheme are validated by numerical tests.