首页|An efficient multi-level high-order algorithm for simulation of a class of Allen-Cahn stochastic systems
An efficient multi-level high-order algorithm for simulation of a class of Allen-Cahn stochastic systems
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NSTL
Elsevier
Computationally quantifying uncertainties in the mathematical modeling of physical processes is crucial for understanding the errors induced by both numerical approximations of the model and lack of precise input data assumed in the continuous model. Uncertain input parameters in the continuous model are typically treated as random variables, leading to the need to consider solutions of both the continuous and discrete models as stochastic processes. Computing statistical moments of the stochastic processes is an extremely important part of the uncertainty quantification problem. In this work, we consider a class of physical processes that are modeled by the Allen-Cahn (A-C) partial differential equation (PDE) evolutionary system, with uncertainties in the initial state of the evolution and the A-C PDE. We develop a hybrid computational model for the stochastic A-C system to efficiently compute statistical moments of the numerical counterparts of the A-C stochastic processes. The hybrid framework comprises finite element method in-space approximations, high-order digital nets based sampling in high dimensional probability space, and an interplay of discretization parameters in the spatial and stochastic approximations. We demonstrate marked efficiency of the hybrid framework, compared to the standard methods, using two- and three-dimensional in space and high stochastic dimensional A-C example systems. (C) 2021 Elsevier B.V. All rights reserved.
Allen-CahnFinite element methodMonte CarloQuasi Monte CarloMulti-levelEQUATIONMOTION
NSTL
Elsevier
