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Numerical Methods For Hyperbolic Conservation Laws With Stiff Relaxation I. Spurious Solutions
 SIAM J. Sci. Comput
, 1992
"... . We consider the numerical solution of hyperbolic systems of conservation laws with relaxation using a shock capturing finite difference scheme on a fixed, uniform spatial grid. We conjecture that certain a priori criteria insure that the numerical method does not produce spurious solutions as the ..."
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Cited by 55 (2 self)
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. We consider the numerical solution of hyperbolic systems of conservation laws with relaxation using a shock capturing finite difference scheme on a fixed, uniform spatial grid. We conjecture that certain a priori criteria insure that the numerical method does not produce spurious solutions as the relaxation time vanishes. One criterion is that the limits of vanishing relaxation time and vanishing viscosity commute for the viscous regularization of the hyperbolic system. A second criterion is that a certain "subcharacteristic" condition be satisfied by the hyperbolic system. We support our conjecture with analytical and numerical results for a specific example, the solution of generalized Riemann problems of a model system of equations with a fractional step scheme in which Godunov's method is coupled with the backward Euler method. We also apply our ideas to the numerical solution of stiff detonation problems. 1. Introduction. Hyperbolic systems of conservation laws with relaxation ...
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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Cited by 48 (14 self)
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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...
The Cauchy problem for the Euler equations for compressible
 In: Handbook of Mathematical Fluid Dynamics
, 2002
"... Abstract. Some recent developments in the study of the Cauchy problem for the Euler equations for compressible fluids are reviewed. The local and global wellposedness for smooth solutions is presented, and the formation of singularity is exhibited; then the local and global wellposedness for disco ..."
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Cited by 14 (4 self)
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Abstract. Some recent developments in the study of the Cauchy problem for the Euler equations for compressible fluids are reviewed. The local and global wellposedness for smooth solutions is presented, and the formation of singularity is exhibited; then the local and global wellposedness for discontinuous solutions, including the BV theory and the L ∞ theory, is extensively discussed. Some recent developments in the study of the Euler equations with source terms are also reviewed.
Traveling Wave Solutions of Fourth Order PDEs for Image Processing
 SIAM J. Math. Anal
, 2004
"... The authors introduce two nonlinear advection diffusion equations, each of which combines Burgers' convection with a fourth order nonlinear diffusion previously designed for image denoising. One equation uses the L²curvature diminishing diffusion of You and Kaveh (IEEE Trans. Image Process., O ..."
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Cited by 14 (5 self)
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The authors introduce two nonlinear advection diffusion equations, each of which combines Burgers' convection with a fourth order nonlinear diffusion previously designed for image denoising. One equation uses the L²curvature diminishing diffusion of You and Kaveh (IEEE Trans. Image Process., October 2000), and the other uses the `Low Curvature Image Simplifiers' diffusion of Tumblin and Turk (SIGGRAPH, August 1999). The new PDEs are compared with a third advection diffusion equation that combines Burgers' convection with a second order diffusion recommended by Perona and Malik for denoising and edge detection (IEEE Trans. Pattern Anal. Machine Intell., July,1990). We prove results regarding the existence and nonexistence of traveling wave solutions of each PDE. Visualizations of each ODE's phase space show qualitative differences between the two fourth order problems. The combined work gives insight into the existence of finite time singularities in solutions of the diffusion equations.
Numerical and Analytical Studies of the Dynamics of Gaseous Detonations
 UNIVERSITY OF CALIFORNIA PRESS
, 2001
"... This thesis examines two dynamic parameters of gaseous detonations, critical energy and cell size. The first part is concerned with the direct initiation of gaseous detonations by a blast wave and the associated critical energy. Numerical simulations of the spherically symmetric direct initiation ev ..."
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Cited by 12 (2 self)
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This thesis examines two dynamic parameters of gaseous detonations, critical energy and cell size. The first part is concerned with the direct initiation of gaseous detonations by a blast wave and the associated critical energy. Numerical simulations of the spherically symmetric direct initiation event with a simple chemical reaction model are presented. Local analysis of the computed unsteady reaction zone structure identifies a competition between heat release rate, front curvature and unsteadiness. The primary failure mechanism is found to be unsteadiness in the induction zone arising from the deceleration of the shock front. On this basis, simplifying assumptions are applied to the governing equations, permitting solution of an analytical model for the critical shock decay rate. The local analysis is validated by integration of reaction zone structure equations with detailed chemical kinetics and prescribed unsteadiness. The model is then applied to the global initiation problem to produce an analytical equation for the critical energy. Unlike previous phenomenological models, this equation is not dependent on other experimentally determined parameters. For di#erent fueloxidizer mixtures, it is found to give agreement with experimental data to within an order of magnitude. The second part of the thesis is concerned with the development of improved reaction models for accurate quantitative simulations of detonation cell size and cellular structure. The mechanism reduction method of Intrinsic LowDimensional Manifolds, originally developed for flame calculations, is shown to be a viable option for detonation simulations when coupled with a separate model in the induction zone. The agreement with detailed chemistry calculations of constant volume reactions and onedim...
A stability index for detonation waves in Majda’s model for reacting flow
 Phys. D
"... Abstract. Using Evans function techniques, we develop a stability index for weak and strong detonation waves analogous to that developed for shock waves in [GZ,BSZ], yielding useful necessary conditions for stability. Here, we carry out the analysis in the context of the Majda model, a simplified mo ..."
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Cited by 11 (9 self)
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Abstract. Using Evans function techniques, we develop a stability index for weak and strong detonation waves analogous to that developed for shock waves in [GZ,BSZ], yielding useful necessary conditions for stability. Here, we carry out the analysis in the context of the Majda model, a simplified model for reacting flow; the method is extended to the full Navier–Stokes equations of reacting flow in [Ly,LyZ]. The resulting stability condition is satisfied for all nondegenerate, i.e., spatially exponentially decaying, weak and strong detonations of the Majda model in agreement with numerical experiments of [CMR] and analytical results of [Sz,LY] for a related model of Majda and Rosales. We discuss also the role in the ZND limit of degenerate, subalgebraically decaying weak detonation and (for a modified, “bumptype ” ignition function) deflagration profiles, as discussed in [GS.1–2] for the full equations. Section
Onedimensional stability of viscous strong detonation waves
"... Abstract. Building on Evans function techniques developed to study the stability of viscous shocks, we examine the stability of viscous strong detonation wave solutions of the reacting NavierStokes equations. The primary result, following [1, 17], is the calculation of a stability index whose sign ..."
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Cited by 10 (9 self)
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Abstract. Building on Evans function techniques developed to study the stability of viscous shocks, we examine the stability of viscous strong detonation wave solutions of the reacting NavierStokes equations. The primary result, following [1, 17], is the calculation of a stability index whose sign determines a necessary condition for spectral stability. We show that for an ideal gas this index can be evaluated in the ZND limit of vanishing dissipative effects. Moreover, when the heat of reaction is sufficiently small, we prove that strong detonations are spectrally stable provided the underlying shock is stable. Finally, for completeness, the stability index calculations for the nonreacting NavierStokes equations are included.
Galloping instability of viscous shock waves
 Physica D
"... Motivated by physical and numerical observations of time oscillatory “galloping”, “spinning”, and “cellular ” instabilities of detonation waves, we study Poincaré–Hopf bifurcation of travelingwave solutions of viscous conservation laws. The main difficulty is the absence of a spectral gap between o ..."
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Cited by 10 (8 self)
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Motivated by physical and numerical observations of time oscillatory “galloping”, “spinning”, and “cellular ” instabilities of detonation waves, we study Poincaré–Hopf bifurcation of travelingwave solutions of viscous conservation laws. The main difficulty is the absence of a spectral gap between oscillatory modes and essential spectrum, preventing standard reduction to a finitedimensional center manifold. We overcome this by direct Lyapunov–Schmidt reduction, using detailed pointwise bounds on the linearized solution operator to carry out a nonstandard implicit function construction in the absence of a spectral gap. The key computation is a spacetime stability estimate on the transverse linearized solution operator reminiscent of Duhamel estimates carried out on the full solution operator in the study of nonlinear stability of spectrally stable traveling waves.
The random projection method for hyperbolic conservation laws with stiff reaction terms
 J. Comput. Phys
"... We propose a random projection method for numerical simulations of hyperbolic conservation laws with stiff source terms arising from chemically reactive flows: Ut + F(U)x + G(U)y = 1 ε �(U). In this problem, the chemical time scales may be orders of magnitude faster than the fluid dynamical time sca ..."
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Cited by 8 (2 self)
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We propose a random projection method for numerical simulations of hyperbolic conservation laws with stiff source terms arising from chemically reactive flows: Ut + F(U)x + G(U)y = 1 ε �(U). In this problem, the chemical time scales may be orders of magnitude faster than the fluid dynamical time scales, making the problem numerically stiff. A classic spurious numerical phenomenon, the incorrect propagation speeds of discontinuities, occurs in underresolved numerical solutions. We introduce a random projection method for the reaction term by replacing the ignition temperature with a uniformly distributed random variable. The statistical average of this method corrects the spurious shock speed, as will be proved with a scalar model problem and demonstrated by a wide range of numerical examples in inviscid detonation waves in both one and two space dimensions. c ○ 2000 Academic Press 1.
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 7 (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