Uncertainty Propagation for Systems of Conservation Laws, High Order Stochastic Spectral Methods
Uncertainty Propagation for Systems of Conservation Laws, High Order Stochastic Spectral Methods
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守恒定律系统的不确定性传播，高阶随机谱方法
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
D. Lucor
中科院分区：
文献类型：
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作者：
Gaël Poëtte;B. Després;D. Lucor
随机Galerkin广义多项式混沌方法（sG-gPC）（Wiener，Am. J. Math.60：897-936，1938;卡梅隆（Cameron）和马丁（Martin），《数学年鉴》（Ann.Math.48：385-392，1947）; Xiu和Karniadakis，SIAM J. Sci. Comp.24（2）：619-644，2002）已知通过非线性守恒律系统（SLC）的不确定性传播遇到若干困难：维度（参见，例如，Nyndham等人，SIAM J. Numer. Anal. 46（5）：2309-2345，2008; Blatman和Sudret，C. R.麦克336：518-523，2008; Witteveen和Bijl，Comp.Struct.86（23-24）：2123-2140，2008）、非线性（参见，例如，Debusshere等人，J. Sci. Comp.26：698-719，2004; Witteveen和Bijl，47 th AIAA Aerospace Sciences Meeting and Exhibit，2006-2066，2006）、不连续性（参见Wan和Karniadakis，SIAM J. Sci. Comp.27（1-3），2006; Lin等人，J.Comp.Phys.217：260-276，2006; Le Maitre和Pillo，J.Comp.Phys.197：28-57，2004; Le Maitre等人，197：502-531，2004; Abgrall，Rapport de Recherche INRIA，2007）。本文首先以一个简单的SLC（p-系统）为例说明在处理非线性和不连续性时所遇到的困难。然后，我们将提出一个新的非自适应高阶不确定性传播方法的基础上的熵的守恒律系统，有效的非线性系统和间断的解决方案。收敛性测试和频谱收敛达到。
The application of the stochastic Galerkin-generalized Polynomial Chaos approach (sG-gPC) (Wiener, Am. J. Math. 60:897–936, 1938; Cameron and Martin, Ann. Math. 48:385–392, 1947; Xiu and Karniadakis, SIAM J. Sci. Comp. 24(2):619–644, 2002) for Uncertainty Propagation through NonLinear Systems of Conservation Laws (SLC) is known to encounter several difficulties: dimensionality (see, e.g., Nobile et al., SIAM J. Numer. Anal. 46(5):2309–2345, 2008; Blatman and Sudret, C. R. Mec. 336:518–523, 2008; Witteveen and Bijl, Comp. Struct. 86(23–24):2123–2140, 2008), non linearities (see, e.g., Debusshere et al., J. Sci. Comp. 26:698–719, 2004; Witteveen and Bijl, 47th AIAA Aerospace Sciences Meeting and Exhibit, 2006–2066, 2006), discontinuities (see Wan and Karniadakis, SIAM J. Sci. Comp. 27(1–3), 2006; Lin et al., J. Comp. Phys. 217:260–276, 2006; Le Maitre and Knio, J. Comp. Phys. 197:28–57, 2004; Le Maitre et al., J. Comp. Phys. 197:502–531, 2004; Abgrall, Rapport de Recherche INRIA, 2007). In this paper, we first illustrate on a simple SLC (p-system) the difficulties occuring when dealing with non linearities and discontinuities. We will then present a new non adaptive high order uncertainty propagation method based on the entropy of the system of conservation laws, efficient on NonLinear systems and discontinuous solutions. Convergence tests are performed and spectral convergence is reached.