The study of advection–diffusion equation has relatively became an active research topic in the field of uncertainty quantification (UQ) due to its numerous real life applications. In this paper, Hermite polynomial chaos is united with summation-by-parts (SBP) – simultaneous approximation terms (SATs) technique to solve the advection–diffusion equations with random Dirichlet boundary conditions (BCs). Polynomial chaos expansion (PCE) with Hermite basis is employed to separate the randomness, then SBP operators are used to approximate the differential operators and SATs are used to enforce BCs by ensuring the stability. For each chaos coefficient, time integration is performed using Runge–Kutta method of fourth order (RK4). Statistical moments namely mean and variance are computed using polynomial chaos coefficients without any extra computational effort. The method is applied on three test problems for validation. The first two test problems are stochastic advection equations on [Formula: see text] without any boundary and third problem is stochastic advection–diffusion equation on [0,2] with Dirichlet BCs. In case of third problem, we have obtained a range of permissible parameters for a stable numerical solution.

中文翻译：

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随机平流-扩散方程的混合 Hermite 多项式混沌 SBP-SAT 技术

对流-扩散方程的研究由于其众多的实际应用，相对成为不确定性量化（UQ）领域的一个活跃研究课题。在本文中，Hermite 多项式混沌与逐部分求和 (SBP) - 同时近似项 (SAT) 技术相结合，以求解具有随机 Dirichlet 边界条件 (BC) 的对流-扩散方程。使用 Hermite 基的多项式混沌扩展 (PCE) 来分离随机性，然后使用 SBP 算子逼近微分算子，并使用 SAT 来确保稳定性来强制执行 BC。对于每个混沌系数，使用四阶龙格-库塔方法 (RK4) 进行时间积分。使用多项式混沌系数计算统计矩，即均值和方差，无需任何额外的计算工作。该方法应用于三个测试问题进行验证。前两个测试问题是 [公式：见文本] 上没有任何边界的随机平流方程，第三个问题是 [0,2] 上使用 Dirichlet BC 的随机平流 - 扩散方程。在第三个问题的情况下，我们已经获得了一个稳定数值解的允许参数范围。