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Macroscopic fluid models with localized kinetic upscaling effects, Multiscale Model
 Simul
"... Abstract. This paper presents a general methodology to design macroscopic fluid models that take into account localized kinetic upscaling effects. The fluid models are solved in the whole domain together with a localized kinetic upscaling that corrects the fluid model wherever it is necessary. This ..."
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Abstract. This paper presents a general methodology to design macroscopic fluid models that take into account localized kinetic upscaling effects. The fluid models are solved in the whole domain together with a localized kinetic upscaling that corrects the fluid model wherever it is necessary. This upscaling is obtained by solving a kinetic equation on the nonequilibrium part of the distribution function. This equation is solved only locally and is related to the fluid equation through a downscaling effect. The method does not need to find an interface condition as do usual domain decomposition methods to match fluid and kinetic representations. We show our approach applies to problems that have a hydrodynamic time scale as well as to problems with diffusion time scale. Simple numerical schemes are proposed to discretized our models, and several numerical examples are used to validate the method. Key words. KineticFluid coupling, Kinetic equation, Hydrodynamic approximation, Diffusion approximation
Diffusion approximation
"... Abstract. This paper presents a general methodology to design macroscopic fluid models that take into account localized kinetic upscaling effects. The fluid models are solved in the whole domain together with a localized kinetic upscaling that corrects the fluid model wherever it is necessary. This ..."
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Abstract. This paper presents a general methodology to design macroscopic fluid models that take into account localized kinetic upscaling effects. The fluid models are solved in the whole domain together with a localized kinetic upscaling that corrects the fluid model wherever it is necessary. This upscaling is obtained by solving a kinetic equation on the nonequilibrium part of the distribution function. This equation is solved only locally and is related to the fluid equation through a downscaling effect. The method does not need to find an interface condition as do usual domain decomposition methods to match fluid and kinetic representations. We show our approach applies to problems that have a hydrodynamic time scale as well as to problems with diffusion time scale. Simple numerical schemes are proposed to discretized our models, and several numerical examples are used to validate the method.
Journal of Computational Physics 180, 120154 (2002)
 J. Comput. Phys
, 1998
"... this paper we discuss the derivation and use of local pressure boundary conditions for finite difference schemes for the unsteady incompressible NavierStokes equations in the velocitypressure formulation. Their use is especially well suited for the computation of moderate to large Reynolds nu ..."
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this paper we discuss the derivation and use of local pressure boundary conditions for finite difference schemes for the unsteady incompressible NavierStokes equations in the velocitypressure formulation. Their use is especially well suited for the computation of moderate to large Reynolds number flows. We explore the similarities between the implementation and use of local pressure boundary conditions and local vorticity boundary conditions in the design of numerical schemes for incompressible flow in 2D. In their respective formulations, when these local numerical boundary conditions are coupled with a fully explicit convectively stable time stepping procedure, the resulting methods are simple to implement and highly efficient. Unlike the vorticity formulation, the use of the local pressure boundary condition approach is readily applicable to 3D flows. The simplicity of the local pressure boundary condition approach and its easy adaptation to more general flow settings make the resulting scheme an attractive alternative to the more popular methods for solving the NavierStokes equations in the velocity pressure formulation. We present numerical results of a secondorder finite difference scheme on a nonstaggered grid using local pressure boundary conditions. Stability and accuracy of the scheme applied to Stokes flow is demonstrated using normal mode analysis. Also described is the extension of the method to variable density flows. c Key Words: incompressible flow; finite difference methods; pressure Poisson solver; local pressure boundary conditions
compressible NavierStokes asymptotics
, 2007
"... stable numerical schemes for the Boltzmann equation preserving the ..."
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AERONAUTICS AND ASTRONAUTICS
"... Redistributed by Stanford University under license with the author. ..."
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Highorder Istable centered difference schemes for viscous compressible flows
"... In this paper we present highorder Istable centered difference schemes for the numerical simulation of viscous compressible flows. Here Istability refers to time discretizations whose linear stability regions contain part of the imaginary axis. This class of schemes has a numerical stability inde ..."
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In this paper we present highorder Istable centered difference schemes for the numerical simulation of viscous compressible flows. Here Istability refers to time discretizations whose linear stability regions contain part of the imaginary axis. This class of schemes has a numerical stability independent of the cellReynolds number Rc, thus allows one to simulate high Reynolds number flows with relatively larger Rc, or coarser grids for a fixed Rc. On the other hand, Rc cannot be arbitrarily large if one tries to obtain adequate numerical resolution of the viscous behavior. We investigate the behavior of highorder Istable schemes for Burgers ’ equation and the compressible NavierStokes equations. We demonstrate that, for the second order scheme, Rc ≤ 3 is an appropriate constraint for numerical resolution of the viscous profile, while for the fourthorder schemes the constraint can be relaxed to Rc ≤ 6. Our study indicates that the fourth order scheme is preferable: better accuracy, higher resolution, and larger cellReynolds numbers. 1
ABSTRACT Title of dissertation: NONLINEAR EVOLUTIONARY PDEs IN IMAGE PROCESSING AND COMPUTER VISION
"... Evolutionary PDEbased methods are widely used in image processing and computer vision. For many of these evolutionary PDEs, there is little or no theory on the existence and regularity of solutions, thus there is little or no understanding on how to implement them effectively to produce the desired ..."
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Evolutionary PDEbased methods are widely used in image processing and computer vision. For many of these evolutionary PDEs, there is little or no theory on the existence and regularity of solutions, thus there is little or no understanding on how to implement them effectively to produce the desired effects. In this thesis work, we study one class of evolutionary PDEs which appear in the literature and are highly degenerate. The study of such second order parabolic PDEs has been carried out by using semigroup theory and maximum monotone operator in case that the initial value is in the space of functions of bounded variation. But the noisy initial image is usually not in this space, it is desirable to know the solution property under weaker assumption on initial image. Following the study of time dependent minimal surface problem, we study the existence and uniqueness of generalized solutions of a class of second order parabolic PDEs. Second order evolutionary PDEbased methods preserve edges very well but sometimes they have undesirable staircase effect. In order to overcome this drawback, fourth order evolu