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RungeKutta Methods for Hyperbolic Conservation Laws with Stiff Relaxation Terms
 J. Comput. Phys
, 1995
"... Underresolved numerical schemes for hyperbolic conservation laws with stiff relaxation terms may generate unphysical spurious numerical results or reduce to lower order if the small relaxation time is not temporally wellresolved. We design a second order RungeKutta type splitting method that posse ..."
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Underresolved numerical schemes for hyperbolic conservation laws with stiff relaxation terms may generate unphysical spurious numerical results or reduce to lower order if the small relaxation time is not temporally wellresolved. We design a second order RungeKutta type splitting method that possesses the discrete analogue of the continuous asymptotic limit, thus is able to capture the correct physical behaviors with high order accuracy even if the initial layer and the small relaxation time are not numerically resolved. Key words. Hyperbolic conservation laws with stiff relaxation, shock capturing difference method, RungeKutta methods, asymptotic limit AMS(MOS) subject classifications. 35L65, 35B40, 65M60 Typeset by A M ST E X 2 1. Introduction Hyperbolic systems with relaxations occur in the study of a variety of physical phenomena, for example in linear and nonlinear waves [42,36], in relaxing gas flow with thermal and chemical nonequilibrium [41,9], in kinetic theory of ra...
Explicit CharacteristicBased HighResolution Algorithms For Hyperbolic Conservation Laws With Stiff Source Terms
, 1996
"... o TA a class while simultaneously taking it for credit. More importantly, for being an extremely valuable mentor, taking special care to introduce me to his colleagues. Further, it was he who provided the initial impetus for the work in Chapter V. 1 Whatever you do, do well. Even if you become a c ..."
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Cited by 8 (0 self)
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o TA a class while simultaneously taking it for credit. More importantly, for being an extremely valuable mentor, taking special care to introduce me to his colleagues. Further, it was he who provided the initial impetus for the work in Chapter V. 1 Whatever you do, do well. Even if you become a crook, just make sure you're a good one. 2 When one of my projects is going nowhere, I leave it (in the magic drawer) and work on a totally different project. When I return and start over, the answers "magically" jump out. iii Thanks to Professors Sichel, Van Leer and Powell for inviting me to Michigan. I have never regretted my decision  hopefully, they have never done so either. My sincere gratitude to Professors Roe, Van Leer, Sichel, Powell and Harabetian, for serving on my committee, for reading through my dissertation at very short notice, and for their valuable insights and comments. Special thanks go, first, to Rosemary, who quickl
A Convex Entropy for a Hyperbolic System with Relaxation
 J. Diff. Eqs
, 1996
"... We explicitly construct a convex entropy function for a hyperbolic system with relaxation. This entropy is defined in the sense of Chen, Levermore and Liu [2], which resembles Boltzman's HTheorem for kinetic equations. This construction follows the idea of Suliciu's energy function for a ..."
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Cited by 7 (0 self)
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We explicitly construct a convex entropy function for a hyperbolic system with relaxation. This entropy is defined in the sense of Chen, Levermore and Liu [2], which resembles Boltzman's HTheorem for kinetic equations. This construction follows the idea of Suliciu's energy function for a ratetype mixed hyperbolicelliptic system [10]. Such an entropy will be useful in proving the entropy property of the relaxation schemes introduced by Jin and Xin for conservation laws [5]. Key words. Hyperbolic conservation laws with relaxation, entropy, HTheorem. Research was supported in part by NSF grant No. DMS9404157. Email address: jin@math.gatech.edu. Typeset by A M ST E X 2 1. Introduction Hyperbolic systems with relaxations occur in the study of a variety of physical phenomena, for example in linear and nonlinear waves [9,12], in relaxing gas flow with thermal and chemical nonequilibrium [3,11], in kinetic theory of rarefied gas dynamics [1], in viscoelasticity [8], multiphase and ...
Simulations of Compressible, Diffusive, Reactive Flows with Detailed Chemistry Using a HighOrder Hybrid WENOCD Scheme
, 2012
"... I would like to firstly thank my advisors Professors Pullin and Shepherd. Without their guidance, support, and patience, this thesis would not have been possible. I would also like to thank everyone else that has worked with me and helped me learn about CFD and detonations/reactive flow research thr ..."
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I would like to firstly thank my advisors Professors Pullin and Shepherd. Without their guidance, support, and patience, this thesis would not have been possible. I would also like to thank everyone else that has worked with me and helped me learn about CFD and detonations/reactive flow research throughout my time at Caltech. Firstly, Dr. Shannon Kao helped me learn about gas dynamics, reactive flow, and taught me CANTERA. Dr. David Hill was the first to help me learn about numerical methods and CFD. I also worked with Dr. Ralf Deiterding at ORNL for one summer and owe much to him for learning how to program with AMROC and parallel computers in general. Ralf also continued to help me throughout the PhD process. Later, I collaborated with Dr. Sally Bane and Mr. Jason Damazo and was able to gain experience in learning how to model and compare to actual experiments. I also owe thanks to Professor Guillaume Blanquart, who helped me learn about and develop a reduced hydrocarbon mechanism.
The Frozen Flow and the Equilibrium Limits in a Polyatomic Gas with Relaxation Effects
"... A travelling waves analysis for a polyatomic gas with n internal energy storage modes is performed. In both the frozen flow and the equilibrium limit the existence and the qualitative behaviour of travelling waves are investigated. Methods from geometric singular perturbation theory are used to stud ..."
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A travelling waves analysis for a polyatomic gas with n internal energy storage modes is performed. In both the frozen flow and the equilibrium limit the existence and the qualitative behaviour of travelling waves are investigated. Methods from geometric singular perturbation theory are used to study these limits. Keywords: frozen flow, equilibrium, shock waves, travelling waves, singular perturbations 1 Introduction We consider a polyatomic gas with n different internal (quadratic) energy storage modes, e.g. vibrational and rotational modes. Many gases like hydrogen, oxygen and mixtures like air have such additional modes. It is known that relaxation mechanisms can affect the internal structure of shock waves and change their thickness [1, 2, 10]. To our knowledge no rigorous results on the existence of such waves have be proven. Our motivation to study this problem stems from the inreased interest in relaxation mechanisms in conservation laws and more specifically in gasdynamic equa...
and
, 1997
"... In this paper we consider a 2_2 relaxation hyperbolic system of conservation laws with a boundary effect, and we show that the solutions of this initial boundary problem tend to the traveling wave solutions of the corresponding Cauchy problem timeasymptotically. In particular, we give the algebraic ..."
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In this paper we consider a 2_2 relaxation hyperbolic system of conservation laws with a boundary effect, and we show that the solutions of this initial boundary problem tend to the traveling wave solutions of the corresponding Cauchy problem timeasymptotically. In particular, we give the algebraic and exponential decay rates by using the weighted energy method. The location of a shift for the traveling wave, to overcome the difficulty in the boundary, plays a key role in this paper.
The Frozen Flow and the Equilibrium Limits in a Polyatomic Gas with Relaxation Effects
, 1998
"... A travelling waves analysis for a polyatomic gas with n internal energy storage modes is performed. In both the frozen flow and the equilibrium limit the existence and the qualitative behaviour of travelling waves are investigated. Methods from geometric singular perturbation theory are used to stud ..."
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A travelling waves analysis for a polyatomic gas with n internal energy storage modes is performed. In both the frozen flow and the equilibrium limit the existence and the qualitative behaviour of travelling waves are investigated. Methods from geometric singular perturbation theory are used to study these limits. Keywords: relaxation effects, frozen flow, equilibrium limit, shock waves, travelling waves, singular perturbations 1 Introduction We consider a polyatomic gas with n different internal (quadratic) energy storage modes, e.g. vibrational and rotational modes. Many gases like hydrogen, oxygen and mixtures like air have such additional modes. It is known that relaxation mechanisms can affect the internal structure of shock waves and change their thickness [1, 2, 11]. To our knowledge no rigorous results on the existence of such waves have be proven. Our motivation to study this problem stems from the increased interest in relaxation mechanisms in conservation laws and more spec...