Results 1  10
of
13
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 ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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.
ON THE ACCURACY OF THE FINITE ELEMENT METHOD PLUS TIME RELAXATION
"... Abstract. If u denotes a local, spatial average of u, thenu ′ = u − u is the associated fluctuation. Consider a time relaxation term added to the usual finite element method. The simplest case for the model advection equation ut + − → a ·∇u = f(x, t) is (uh,t + − → a ·∇uh,vh)+χ(u ′ h,v ′ h)=(f(x, t ..."
Abstract
 Add to MetaCart
Abstract. If u denotes a local, spatial average of u, thenu ′ = u − u is the associated fluctuation. Consider a time relaxation term added to the usual finite element method. The simplest case for the model advection equation ut + − → a ·∇u = f(x, t) is (uh,t + − → a ·∇uh,vh)+χ(u ′ h,v ′ h)=(f(x, t),vh). We analyze the error in this and (more importantly) higher order extensions and show that the added time relaxation term not only suppresses excess energy in marginally resolved scales but also increases the accuracy of the resulting finite element approximation. 1.
A SIMILARITY THEORY OF APPROXIMATE DECONVOLUTION MODELS OF TURBULENCE
"... We apply the phenomenology of homogeneous, isotropic turbulence to the family of approximate deconvolution models proposed by Stolz and Adams. In particular, we establish that the models themselves have an energy cascade with two asymptotically different inertial ranges. Delineation of these gives ..."
Abstract
 Add to MetaCart
We apply the phenomenology of homogeneous, isotropic turbulence to the family of approximate deconvolution models proposed by Stolz and Adams. In particular, we establish that the models themselves have an energy cascade with two asymptotically different inertial ranges. Delineation of these gives insight into the resolution requirements of using approximate deconvolution models. The approximate deconvolution model’s energy balance contains both an enhanced energy dissipation and a modi…cation to the model’s kinetic energy. The modification of the model’s kinetic energy induces a secondary energy cascade which accelerates scale truncation. The enhanced energy dissipation completes the scale truncation by reducing the model’s microscale from the Kolmogorov microscale.
Bounds on Energy and Helicity Dissipation Rates of . . .
"... We consider a family of high accuracy, approximate deconvolution models of turbulence. For body force driven turbulence, we prove directly from the models equations of motion the following bounds on the model’s time averaged energy dissipation rate, < "ADM>, and helicity dissipation rate, < ADM (w) ..."
Abstract
 Add to MetaCart
We consider a family of high accuracy, approximate deconvolution models of turbulence. For body force driven turbulence, we prove directly from the models equations of motion the following bounds on the model’s time averaged energy dissipation rate, < "ADM>, and helicity dissipation rate, < ADM (w)>;
Copies of this report are available from:
, 2002
"... CSE curricula at the ETH zürich on the internet: ..."