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Diffusive BGK Approximations for Nonlinear Multidimensional Parabolic Equations
, 1999
"... We introduce a class of discrete velocity BGK type approximations to multidimensional scalar nonlinearly diffusive conservation laws. We prove the wellposedness of these models, a priori bounds and kinetic entropy inequalities that allow to pass into the limit towards the unique entropy solution re ..."
Abstract

Cited by 24 (3 self)
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We introduce a class of discrete velocity BGK type approximations to multidimensional scalar nonlinearly diffusive conservation laws. We prove the wellposedness of these models, a priori bounds and kinetic entropy inequalities that allow to pass into the limit towards the unique entropy solution recently obtained by Carrillo. Examples of such BGK models are provided.
CharacteristicBased Numerical Schemes for Hyperbolic Systems With Nonlinear Relaxation
, 1997
"... In order to embark the development of characteristic based schemes for hyperbolic systems with nonlinear stiff source terms we have studied a prototype onedimensional discretevelocity Boltzmann equation. We show that the method can be evaluated at the cost of an explicit scheme and that yield accu ..."
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Cited by 4 (3 self)
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In order to embark the development of characteristic based schemes for hyperbolic systems with nonlinear stiff source terms we have studied a prototype onedimensional discretevelocity Boltzmann equation. We show that the method can be evaluated at the cost of an explicit scheme and that yield accurate solutions even when the relaxation time is much less than the time step. Numerical experiments confirm that the quality of the results obtained is comparable with those of earlier approaches. 1 Introduction This paper is concerned with the development of a method of characteristics for hyperbolic systems with nonlinear relaxation [5]. These systems are characterized by the presence of multiple scales in the problem depending on the relaxation time ffl. The flow starts out at the frozen limit (t=ffl ! 0) and relaxes to the equilibrium limit (t=ffl ! 1). When at least one of these scales is much smaller than the others, the system is said to be stiff. The most popular way to solve numeri...
LongTime Diffusive Behavior of Solutions to a Hyperbolic Relaxation System
 Asymptot. Anal
, 2001
"... We study the large time behavior of the solutions to the Cauchy problem for the system with relaxation source with (x; t) 2 R \Theta R , for the initial data (u; v) = (u 0 ; v 0 ) at t = 0, with u 0 ; v 0 2 L (R)"L (R), f(u) = ffu =2 and jv 0 j au 0 . Under the subcharacteristic condit ..."
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Cited by 4 (3 self)
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We study the large time behavior of the solutions to the Cauchy problem for the system with relaxation source with (x; t) 2 R \Theta R , for the initial data (u; v) = (u 0 ; v 0 ) at t = 0, with u 0 ; v 0 2 L (R)"L (R), f(u) = ffu =2 and jv 0 j au 0 . Under the subcharacteristic condition we show that, as t ! 1, the component u tends towards a fundamental solution of the convectiondiffusion equation u xx in the L norm, at a rate faster than t \Gamma(p\Gamma1)=2p .
Investigation of the discontinuous Galerkin method for firstorder PDE approaches to CFD
 17th AIAA Computational Flow Dynamics Conference
, 2005
"... To simulate accurately and efficiently aerospacetype flows using solely firstorder PDE models, new numerical methods may be required to overcome difficulties introduced by the model. We consider a Discontinuous Galerkin (DG) method that has been shown to possess an asymptotic preservation (AP) pro ..."
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Cited by 3 (1 self)
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To simulate accurately and efficiently aerospacetype flows using solely firstorder PDE models, new numerical methods may be required to overcome difficulties introduced by the model. We consider a Discontinuous Galerkin (DG) method that has been shown to possess an asymptotic preservation (AP) property for onedimensional hyperbolic relaxation systems, in an effort to understand the source of this property and to determine if this property persists for problems in which we are interested. Comparison with traditional MUSCL schemes suggests that the source of the known AP property results from a coupling of the flux and source terms by the direct evolution of the solution slope. For a canonical model problem in a single spatial dimension, we use Von Neumann analysis to demonstrate that the limiting flux function of the DG scheme is of the HartenLaxVan Leer type. Further Von Neumann analysis of both a MUSCL scheme and the DG scheme show that near equilibrium, both methods have mesh size restrictions related to the resolution of physical dissipation. The DG scheme is less numerically dissipative than the HR scheme in this limit. Numerical simulations demonstrate these findings in one dimension. In one and two dimensions, our numerical results demonstrate that the convergence rate of the DG scheme eventually drops as the mesh is refined.