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The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations
tl;dr
We present a new method for solving stochastic differential equations based on Galerkin projections and extensions of Wiener's polynomial chaos that reduces the dimensionality of the system and leads to exponential convergence. Expand
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Spectral/hp Element Methods for Computational Fluid Dynamics
Introduction Fundamental concepts in one dimension Multi-dimensional expansion bases Multi-dimensional formulations Diffusion equation Advection and advection-diffusion Non-conforming elementsExpand
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High-order splitting methods for the incompressible Navier-Stokes equations
Abstract A new pressure formulation for splitting methods is developed that results in high-order time-accurate schemes for the solution of the incompressible Navier-Stokes equations. In particular,Expand
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REPORT: A MODEL FOR FLOWS IN CHANNELS, PIPES, AND DUCTS AT MICRO AND NANO SCALES
Rarefied gas flows in channels, pipes, and ducts with smooth surfaces are studied in a wide range of Knudsen number (Kn) at low Mach number (M) with the objective of developing simple, physics-basedExpand
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Spectral/hp Element Methods for CFD
Introduction Fundamental concepts in one dimension Multi-dimensional expansion bases Multi-dimensional formulations Diffusion equation Advection and advection-diffusion Non-conforming elementsExpand
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Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
tl;dr
We introduce physics-informed neural networks that are trained to solve supervised learning tasks while respecting any given laws of physics described by general nonlinear partial differential equations. Expand
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Modeling uncertainty in flow simulations via generalized polynomial chaos
We present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations. The stochastic input is represented spectrally by employing orthogonal polynomialExpand
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  • Open Access
An adaptive multi-element generalized polynomial chaos method for stochastic differential equations
We formulate a Multi-Element generalized Polynomial Chaos (ME-gPC) method to deal with long-term integration and discontinuities in stochastic differential equations. We first present this method forExpand
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Dynamics and low-dimensionality of a turbulent near wake
We investigate the dynamics of the near wake in turbulent flow past a circular cylinder up to ten cylinder diameters downstream. The very near wake (up to three diameters) is dominated by the shearExpand
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The Development of Discontinuous Galerkin Methods
tl;dr
In this paper we present an overview of the evolution of the discontinuous Galerkin methods since their introduction in 1973 by Reed and Hill, in the framework of neutron transport, until their most recent developments. Expand
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