A Study of The Stochastic Burgers’ Equation Using The Dynamical Orthogonal Method

Abstract

In the current work, the stochastic Burgers’ equation is studied using the Dynamically Orthogonal (DO) method. The DO presents a low-dimensional representation for the stochastic fields. Unlike many other methods, it has a time-dependent property on both the spatial basis and stochastic coefficients, with flexible representation especially in the strongly transient and nonstationary problems. We consider a computational study and application of the DO method and compare it with the Polynomial Chaos (PC) method. For comparison, both the stochastic viscous and inviscid Burgers’ equations are considered. A hybrid approach, combining the DO and PC is proposed in case of deterministic initial conditions to overcome the singularities that occur in the DO method. The results are verified with the stochastic collocation method. Overall, we observe that the DO method has a higher rate of convergence as the number of modes increases. The DO method is found to be more efficient than PC for the same level of accuracy, especially for the case of high-dimensional parametric spaces. The inviscid Burgers’ equation is analyzed to study the shock wave formation when using the DO after suitable handling of the convective term. The results show that the sinusoidal wave shape is distorted and sharpened as the time evolves till the shock wave occurs.