A convergence study for SPDEs using combined Polynomial Chaos and Dynamically-Orthogonal schemes

作者：Choi Minseok; Sapsis Themistoklis P; Karniadakis George Em^*

来源：Journal of Computational Physics, 2013, 245: 281-301.

DOI：10.1016/j.jcp.2013.02.047

• Stochastic partial differential equations
• Uncertainty quantification
• High dimensions
• Stochastic collocation
• Low-dimensionality

We study the convergence properties of the recently developed Dynamically Orthogonal (DO) field equations [1] in comparison with the Polynomial Chaos (PC) method. To this end, we consider a series of one-dimensional prototype SPDEs, whose solution can be expressed analytically, and which are associated with both linear (advection equation) and nonlinear (Burgers equation) problems with excitations that lead to unimodal and strongly bi-modal distributions. We also propose a hybrid approach to tackle the singular limit of the DO equations for the case of deterministic initial conditions. The results reveal that the DO method converges exponentially fast with respect to the number of modes (for the problems considered) giving same levels of computational accuracy comparable with the PC method but (in many cases) with substantially smaller computational cost compared to stochastic collocation, especially when the involved parametric space is high-dimensional.

• 出版日期2013-7-15