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首页> 外文期刊>Journal of Computational Physics >Numerical analysis of the Burgers' equation in the presence of uncertainty
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Numerical analysis of the Burgers' equation in the presence of uncertainty

机译：存在不确定性时Burgers方程的数值分析

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摘要

The Burgers' equation with uncertain initial and boundary conditions is investigated using a polynomial chaos (PC) expansion approach where the solution is represented as a truncated series of stochastic, orthogonal polynomials. The analysis of well-posedness for the system resulting after Galerkin projection is presented and follows the pattern of the corresponding deterministic Burgers equation. The numerical discretization is based on spatial derivative operators satisfying the summation by parts property and weak boundary conditions to ensure stability. Similarly to the deterministic case, the explicit time step for the hyperbolic stochastic problem is proportional to the inverse of the largest eigenvalue of the system matrix. The time step naturally decreases compared to the deterministic case since the spectral radius of the continuous problem grows with the number of polynomial chaos coefficients. An estimate of the eigenvalues is provided. A characteristic analysis of the truncated PC system is presented and gives a qualitative description of the development of the system over time for different initial and boundary conditions. It is shown that a precise statistical characterization of the input uncertainty is required and partial information, e.g. the expected values and the variance, are not sufficient to obtain a solution. An analytical solution is derived and the coefficients of the infinite PC expansion are shown to be smooth, while the corresponding coefficients of the truncated expansion are discontinuous.
机译：使用多项式混沌（PC）展开方法研究了具有不确定初始条件和边界条件的Burgers方程，其中，解决方案表示为随机正交正交多项式的截断序列。给出了在Galerkin投影之后得到的系统的适定性分析，并遵循相应的确定性Burgers方程的模式。数值离散化基于空间导数算子，该空间导数算子满足零件性质和弱边界条件的总和，以确保稳定性。类似于确定性情况，双曲型随机问题的显式时间步长与系统矩阵的最大特征值的倒数成比例。与确定性情况相比，时间步长自然减少了，因为连续问题的频谱半径随多项式混沌系数的数量而增长。提供了特征值的估计。提出了截短的PC系统的特性分析，并给出了针对不同初始条件和边界条件随时间推移系统发展的定性描述。结果表明，需要对输入不确定度进行精确的统计表征，并且需要部分信息，例如期望值和方差不足以获得解决方案。推导了一个解析解，无限PC展开的系数被证明是平滑的，而相应的截断展开的系数是不连续的。

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