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Numerical Analysis of a Higher Order Time Relaxation Model of Fluids
"... We study the numerical errors in finite element discretizations of a time relaxation model of fluid motion: ut + u · ∇u + ∇p − ν∆u + χu ∗ = f and ∇ · u = 0 In this model, introduced by Stolz, Adams and Kleiser, u ∗ is a generalized fluctuation and χ the time relaxation parameter. The goal of inclu ..."
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We study the numerical errors in finite element discretizations of a time relaxation model of fluid motion: ut + u · ∇u + ∇p − ν∆u + χu ∗ = f and ∇ · u = 0 In this model, introduced by Stolz, Adams and Kleiser, u ∗ is a generalized fluctuation and χ the time relaxation parameter. The goal of inclusion of the χu ∗ is to drive unresolved fluctuations to zero exponentially. We study convergence of discretization of the model to the model’s solution as h, ∆t → 0. Next we complement this with an experimental study of the effect the time relaxation term (and a nonlinear extension of it) has on the large scales of a flow near a transitional point. We close by showing that the time relaxation term does not alter shock speeds in the inviscid, compressible case, giving analytical confirmation of a result of Stolz, Adams and Kleiser.
NUMERICAL ANALYSIS OF FILTER BASED STABILIZATION FOR EVOLUTION EQUATIONS
"... Abstract. We consider filter based stabilization for evolution equations (in general) and for the NavierStokes equations (in particular). The first method we consider is to advance in time one time step by a given method and then to apply an (uncoupled and modular) filter to get the approximation a ..."
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Abstract. We consider filter based stabilization for evolution equations (in general) and for the NavierStokes equations (in particular). The first method we consider is to advance in time one time step by a given method and then to apply an (uncoupled and modular) filter to get the approximation at the new time level. This filter based stabilization, although algorithmically appealing, is viewed in the literature as introducing far too much numerical dissipation to achieve a quality approximate solution. We show that this is indeed the case. We then consider a modification: Evolve one time step, Filter, Deconvolve then Relax to get the approximation at the new time step. We give a precise analysis of the numerical diffusion and error in this process and show it has great promise, confirmed in several numerical experiments. 1. Introduction. Simulations
Dimension reduction method for ODE fluid models
 J. Comp. Phys
"... We develop a new dimension reduction method for large size ODE systems obtained from a discretization of partial differential equations of viscous single and multiphase fluid flow. The method is also applicable to other large size classical particle systems with negligibly small variations of partic ..."
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We develop a new dimension reduction method for large size ODE systems obtained from a discretization of partial differential equations of viscous single and multiphase fluid flow. The method is also applicable to other large size classical particle systems with negligibly small variations of particle concentration. We propose a new computational closure for mesoscale balance equations based on numerical iterative deconvolution. To illustrate the computational advantages of the proposed reduction method, we use it to solve a system of Smoothed Particle Hydrodynamic ODEs describing single phase and twophase layered Poiseuille flows driven by uniform and periodic (in space) body forces. For the single phase Poiseuille flow driven by the uniform force, the coarse solution was obtained with the zeroorder deconvolution. For the single phase flow driven by the periodic body force and for the twophase flows, the higherorder (the first and secondorder) deconvolutions were necessary to obtain a sufficiently accurate solution. Key words: model reduction, ODEs, multiscale modeling, coarse integration, upscaling, closure problem, deconvolution Preprint 1
DECONVOLUTION CLOSURE FOR MESOSCOPIC CONTINUUM MODELS OF PARTICLE SYSTEMS
"... Abstract. The paper introduces a general framework for derivation of continuum equations governing mesoscale dynamics of large particle systems. The balance equations for spatial averages such as density, linear momentum, and energy were previously derived by a number of authors. These equations ar ..."
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Abstract. The paper introduces a general framework for derivation of continuum equations governing mesoscale dynamics of large particle systems. The balance equations for spatial averages such as density, linear momentum, and energy were previously derived by a number of authors. These equations are not in closed form because the stress and the heat flux cannot be evaluated without the knowledge of particle positions and velocities. We propose a closure method for approximating fluxes in terms of other mesoscale averages. The main idea is to rewrite the nonlinear averages as linear convolutions that relate microand mesoscale dynamical functions. The convolutions can be approximately inverted using regularization methods developed for solving illposed problems. This yields closed form constitutive equations that can be evaluated without solving the underlying ODEs. We test the method numerically on FermiPastaUlam chains with two different potentials: the classical LennardJones, and the purely repulsive potential used in granular materials modeling. The initial conditions incorporate velocity fluctuations on scales that are smaller than the size of the averaging window. The results show very good agreement between the exact stress and its closed form approximation.
SUPERCONVERGENCE OF FINITE ELEMENT DISCRETIZATION OF TIME RELAXATION MODELS OF ADVECTION
"... Abstract. The nodal accuracy of …nite element discretizations of advection equations including a time relaxation term is studied. Worst case error estimates have been proven for this combination by energy methods. By considering the Cauchy problem with uniform meshes, precise Fourier analysis of the ..."
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Abstract. The nodal accuracy of …nite element discretizations of advection equations including a time relaxation term is studied. Worst case error estimates have been proven for this combination by energy methods. By considering the Cauchy problem with uniform meshes, precise Fourier analysis of the error is possible. This analysis shows (1)the worst case upper bounds are sharp, (2)time relaxation stabilization does not degrade superconvergence of the usual FEM, and (3)higher order time relaxation is preferable to maintain small numerical errors. Key words. superconvergence, time relaxation, deconvolution 1. Introduction. We
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, 2002
"... CSE curricula at the ETH zürich on the internet: ..."
Journal of Scientific Computing ( © 2006) DOI: 10.1007/s1091500590299 Spectral Vanishing Viscosity Method for LargeEddy Simulation of Turbulent Flows
, 2004
"... An efficient spectral vanishing viscosity method for the largeeddy simulation of incompressible flows is proposed, both for standard spectral and spectral element approximations. The approach is integrated in a collocation spectral ChebyshevFourier solver and then used to compute the turbulent wak ..."
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An efficient spectral vanishing viscosity method for the largeeddy simulation of incompressible flows is proposed, both for standard spectral and spectral element approximations. The approach is integrated in a collocation spectral ChebyshevFourier solver and then used to compute the turbulent wake of a cylinder in a crossflow confined geometry (Reynolds number Re = 3900).
DISCRETE MODELS OF FLUIDS: SPATIAL AVERAGING, CLOSURE AND MODEL REDUCTION
"... Abstract. The main question addressed in the paper is how to obtain closed form continuum equations governing spatially averaged dynamics of semidiscrete ODE models of fluid flow. In the presence of multiple small scale heterogeneities, the size of these ODE systems can be very large. Spatial avera ..."
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Abstract. The main question addressed in the paper is how to obtain closed form continuum equations governing spatially averaged dynamics of semidiscrete ODE models of fluid flow. In the presence of multiple small scale heterogeneities, the size of these ODE systems can be very large. Spatial averaging is then a useful tool for reducing computational complexity of the problem. The averages satisfy balance equations of mass, momentum and energy. These equations are exact, but they do not form a continuum model in the true sense of the word because calculation of stress and heat flux requires solving the underlying ODE system. To produce continuum equations that can be simulated without resolving microscale dynamics, we developed a closure method based on the use of regularized deconvolutions. We mostly deal with nonlinear averaging suitable for Lagrangian particle solvers, but consider Eulerian linear averaging where appropriate. The results of numerical experiments show good agreement between our closed form flux approximations and their exact counterparts.
Journal of Mathematical Fluid Mechanics Mathematical Perspectives on Large Eddy Simulation Mod els for Turbulent Flows
"... Abstract. The main objective of this paper is to review and report on key mathematical issues related to the theory of Large Eddy Simulation of turbulent flows. We review several LES models for which we attempt to provide mathematical justifications. For instance, some filtering techniques and nonl ..."
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Abstract. The main objective of this paper is to review and report on key mathematical issues related to the theory of Large Eddy Simulation of turbulent flows. We review several LES models for which we attempt to provide mathematical justifications. For instance, some filtering techniques and nonlinear viscosity models are found to be regularization techniques that transform the possibly illposed Navier–Stokes equation into a wellposed set of PDE’s. Spectral eddyviscosity methods are also considered. We show that these methods are not spectrally accurate, and, being quasilinear, that they fail to be regularizations of the Navier–Stokes equations. We then propose a new spectral hyperviscosity model that regularizes the Navier–Stokes equations while being spectrally accurate. We finally review scalesimilarity models and twoscale subgrid viscosity models. A new energetically coherent scalesimilarity model is proposed for which the filter does not require any commutation property nor solenoidality of the advection field. We also show that twoscale methods are mathematically justified in the sense that, when applied to linear noncoercive PDE’s, they actually yield convergence in the graph norm.