Upwind Finite-Volume Solution of Stochastic Burgers’ Equation

Abstract

In this paper, a stochastic finite-volume solver based on polynomial chaos expansion is developed. The upwind scheme is used to avoid the numerical instabilities. The Burgers' equation subjected to deterministic boundary conditions and random viscosity is solved. The solution uncertainty is quantified for different values of viscosity. Monte-Carlo simulations are used to validate and compare the developed solver. The mean, standard deviation and the probability distribution function (p.d.f) of the stochastic Burgers' solution is quantified and the effect of some parameters is investigated. The large sparse linear system resulting from the stochastic solver is solved in parallel to enhance the performance. Also, Monte-Carlo simulations are done in parallel and the execution times are compared in both cases.

Stochastic 2D Incompressible Navier-Stokes Solver Using the Vorticity-Stream Function Formulation

“…Assuming that the current system ((1)- (3)) has one or more parameters and/or boundary conditions that have stochastic variations, this will lead to a stochastic system. Following the procedure given in [9,11,15] by expanding all variables and parameters using PCE, (11), substituting in (1), (2), and (3) and taking the ensemble average after multiplying by Ψ ; 0 ≤ ≤ to get the following stochastic system:…”

“…In recent years, the applications of SSFEM in computational fluid dynamics (CFD) are expanded. In [11] a stochastic finite-volume upwind technique is used to solve viscous Burgers' equation with stochastic viscosity over a wide range of the mean viscosity. The stochastic developed solver has higher performance compared with the MC simulations.…”

Dynamically Orthogonal Numerical Schemes for Efficient Stochastic Advection and Lagrangian Transport

“…For model order reduction of SPDEs, classic methods such as polynomial chaos Downloaded 12/12/18 to 18.51.0.96. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php [56,28,84,13], proper orthogonal decomposition (POD) [26,60], dynamic mode decomposition (DMD) [61,66,78,82,31], and stochastic Galerkin schemes and adjoint methods [10,7] assume a priori choices of time-independent modes \bfitu i (\bfitx ) and/or rely on Gaussianity assumptions on the probability distribution of the coefficients \zeta i . For example, the popular data POD [26] and DMD [66] methods suggest extracting timeindependent modes \bfitu i (\bfitx ) that respectively best represent the variability (for the POD method) or the approximate linear dynamics (for the DMD method) of a series of snapshots \bfitu (t k , \bfitx , \omega 0 ) for a given observed or simulated realization \omega 0 .…”

“…How to adapt these schemes for reduced-order numerical advection, which cannot afford examining the realizations individually, is therefore particularly challenging [77,80,65]. This explains in part why many stochastic advection attempts have essentially restricted themselves to 1D applications [19,28,13,56] or simplified 2D cases that do not exhibit strong shocks [81].…”