## Explicit Characteristic-Based High-Resolution Algorithms For Hyperbolic Conservation Laws With Stiff Source Terms (1996)

Citations: | 8 - 0 self |

### BibTeX

@TECHREPORT{Arora96explicitcharacteristic-based,

author = {Mohit Arora},

title = {Explicit Characteristic-Based High-Resolution Algorithms For Hyperbolic Conservation Laws With Stiff Source Terms},

institution = {},

year = {1996}

}

### OpenURL

### Abstract

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

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Citation Context ...emory effects 5 including computational combustion 1 2 a is some characteristic velocity). This dissertation will be restricted to systems of conservation laws with relaxation in the sense of Whitham =-=[78, 79]-=- and Liu [49], in which an N \Theta N hyperbolic system relaxes in the limit ofs! 0 to an M \Theta M system (M ! N ). An additional restriction is that the equilibrium eigenvalues satisfy Liu's sub--c... |

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Citation Context ...king ahead, the eventual goal of this project is to solve non--linear systems with stiff relaxation terms. A natural framework for solving non--linear homogeneous systems is the finite--volume method =-=[46]-=-. Thus, in this chapter, an attempt will be made to come up with a good fully--discrete finite--volume formulation for solving the general linear 2 \Theta 2 system shown in Chapter II. The discretizat... |

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Citation Context ...perbolic systems of conservation laws, where the source terms cause a "relaxation" towards equilibrium. Such systems describe a wide variety of physical phenomena, such as gas flows with rel=-=axation 1 [37, 36, 8, 78, 20, 81, 82, 21, 65, 47, 48, 64, 42, 73, 5, 77, 12, 53, 52]-=-, multiphase flow 2 [32, 51, 28, 75] and phase transitions [68], turbulence [30, 45], water waves 3 [66], viscoelasticity 4 [60] and reactive flows 5 [9, 29, 50, 23]. Further, they govern models for t... |

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Citation Context ...n this dissertation to be of uniform width \Deltax (Figure 3.1). However, the formulation can be extended quite readily to non--uniform meshes, such as those that arise in adaptive--grid computations =-=[6, 26, 59]-=-. In the finite--volume formulation, the cell averaged value in cell 41 j at time level t n (w n j ), defined by v v v j j \Gamma 1 j + 1 t n t n+1 t n+ff 3 t n+ff 2 t n+ff 1 v v v j \Gamma 1=2 j + 1=... |

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Citation Context ...rtially offset by the loss of accuracy, a situation that does not arise in ODE's but does in the context of wave--propagation problems. A third, and frequently used alternative, is operator splitting =-=[67]-=-, in which the hydrodynamic and non--equilibrium equations are solved alternately. While this method appears reasonable, it is probably not the best choice (simply because two physically simultaneous ... |

138 | Hyperbolic conservation laws with stiff relaxation terms and entropy
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Citation Context ...prototypical 2 \Theta 2 system. The stability of this system (under the sub--characteristic condition) and that of its zero relaxation limit (the time--asymptotic solution 7 ) were proven in [49] and =-=[18]-=-, respectively. Relaxation systems are characterized by dispersive wave behavior 8 and the presence of multiple scales in the problem. The flow starts out at the frozen limit (t= ! 0) and relaxes to t... |

133 |
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Citation Context ...ncluding computational combustion 1 2 a is some characteristic velocity). This dissertation will be restricted to systems of conservation laws with relaxation in the sense of Whitham [78, 79] and Liu =-=[49]-=-, in which an N \Theta N hyperbolic system relaxes in the limit ofs! 0 to an M \Theta M system (M ! N ). An additional restriction is that the equilibrium eigenvalues satisfy Liu's sub--characteristic... |

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Citation Context ...perbolic systems of conservation laws, where the source terms cause a "relaxation" towards equilibrium. Such systems describe a wide variety of physical phenomena, such as gas flows with rel=-=axation 1 [37, 36, 8, 78, 20, 81, 82, 21, 65, 47, 48, 64, 42, 73, 5, 77, 12, 53, 52]-=-, multiphase flow 2 [32, 51, 28, 75] and phase transitions [68], turbulence [30, 45], water waves 3 [66], viscoelasticity 4 [60] and reactive flows 5 [9, 29, 50, 23]. Further, they govern models for t... |

79 |
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Citation Context ...ilibrium. Such systems describe a wide variety of physical phenomena, such as gas flows with relaxation 1 [37, 36, 8, 78, 20, 81, 82, 21, 65, 47, 48, 64, 42, 73, 5, 77, 12, 53, 52], multiphase flow 2 =-=[32, 51, 28, 75]-=- and phase transitions [68], turbulence [30, 45], water waves 3 [66], viscoelasticity 4 [60] and reactive flows 5 [9, 29, 50, 23]. Further, they govern models for the kinetic theory of gases [17, 14, ... |

62 | Uniformly accurate schemes for hyperbolic system with relaxation
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Citation Context ...onserved variables density ae (+), momentum m (o) and z (\Pi) are plotted for the Broadwell model (Equation 6.2 ) for four different initial conditions, all of which may be compared to the results in =-=[15]-=- . Each solution is plotted at a final time t = 0:5, and computed with a CFL numbers= 0:5. The data used, the relaxation distancesand the mesh size dx are given below. Initial data A: wL = (1; 1; 1) T... |

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Citation Context ...ilibrium. Such systems describe a wide variety of physical phenomena, such as gas flows with relaxation 1 [37, 36, 8, 78, 20, 81, 82, 21, 65, 47, 48, 64, 42, 73, 5, 77, 12, 53, 52], multiphase flow 2 =-=[32, 51, 28, 75]-=- and phase transitions [68], turbulence [30, 45], water waves 3 [66], viscoelasticity 4 [60] and reactive flows 5 [9, 29, 50, 23]. Further, they govern models for the kinetic theory of gases [17, 14, ... |

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Citation Context ...s such properties is the point-- implicit method [83], where the source term is treated implicitly (locally). Studies focusing on numerical algorithm development for relaxation systems have been many =-=[31, 54, 55, 71]-=-, but only recently have good results been obtained for the difficult fluid dynamic limit by Jin and Levermore [41], Jin [38] and Caflisch, Jin and Russo [15]. It was shown there that poor results are... |

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Citation Context ...eory of gases [17, 14, 7, 11, 33], and applications of these methods to kinetic--based schemes for compressible flow [27, 58, 56, 57] and rarefied--gas flows (including MHD and re--entry hypersonics) =-=[25, 10, 13]-=-. As is clear, such systems form an extremely important class of problems, and are typically of the form w t + f x = \Gamma 1 s ; (1.1) where w; f ; s are state, flux and source vectors respectively, ... |

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Citation Context ....2 ) in the intermediate regime [dx = 0:02sO( ), (+)] ats= 0:5 superposed onto the "exact" solution [dx = 0:001 (solid line)]. : : : 56 4.3 Solution to the very stiff (and strongly non--line=-=ar) cases [38] (also see-=- Section 4.2.3 ). Here, dx = 0:01;s= 10 \Gamma10 ;s= 0:03; \Deltat= ? 10 6 Left Figure: Initial data in local equilibrium, numerical solution (+) superposed onto the "exact" solution [solid ... |

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Citation Context ...n this dissertation to be of uniform width \Deltax (Figure 3.1). However, the formulation can be extended quite readily to non--uniform meshes, such as those that arise in adaptive--grid computations =-=[6, 26, 59]-=-. In the finite--volume formulation, the cell averaged value in cell 41 j at time level t n (w n j ), defined by v v v j j \Gamma 1 j + 1 t n t n+1 t n+ff 3 t n+ff 2 t n+ff 1 v v v j \Gamma 1=2 j + 1=... |

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Citation Context ...h results allowed to degrade gracefully as the stiffness (under--resolution) increases. How9 This is not the first instance of using the asymptotics in numerical schemes, these having been applied in =-=[44, 39, 43, 40]-=-. 5 ever, a natural requirement is that the method result in physical solutions (captured shocks with correct propagation speeds and jumps), even for severely under--resolved cases. The non--relaxable... |

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Citation Context ...82, 21, 65, 47, 48, 64, 42, 73, 5, 77, 12, 53, 52], multiphase flow 2 [32, 51, 28, 75] and phase transitions [68], turbulence [30, 45], water waves 3 [66], viscoelasticity 4 [60] and reactive flows 5 =-=[9, 29, 50, 23]-=-. Further, they govern models for the kinetic theory of gases [17, 14, 7, 11, 33], and applications of these methods to kinetic--based schemes for compressible flow [27, 58, 56, 57] and rarefied--gas ... |

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Citation Context ...)] x = \Gamma [v \Gamma g(u)] : (5.9) Now, let v = g(u) + v 1 ; (5.10) where it is assumed that v 1 (a low frequency component) is small and that its derivativess(i.e., v 1t ; v 1x ) are even smaller =-=[19]-=-. Then, v t = [g(u)] t = g 0 (u) \Delta u t ; (5.11) v x = [g(u)] x = g 0 (u) \Delta u x ; (5.12) and Equations 5.9 and 5.10 give v 1 = \Gammav t \Gamma [p(u)] x ; where [p(u)] x = p 0 (u) \Delta u x ... |

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Citation Context ...s flows with relaxation 1 [37, 36, 8, 78, 20, 81, 82, 21, 65, 47, 48, 64, 42, 73, 5, 77, 12, 53, 52], multiphase flow 2 [32, 51, 28, 75] and phase transitions [68], turbulence [30, 45], water waves 3 =-=[66]-=-, viscoelasticity 4 [60] and reactive flows 5 [9, 29, 50, 23]. Further, they govern models for the kinetic theory of gases [17, 14, 7, 11, 33], and applications of these methods to kinetic--based sche... |

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Citation Context ...h results allowed to degrade gracefully as the stiffness (under--resolution) increases. How9 This is not the first instance of using the asymptotics in numerical schemes, these having been applied in =-=[44, 39, 43, 40]-=-. 5 ever, a natural requirement is that the method result in physical solutions (captured shocks with correct propagation speeds and jumps), even for severely under--resolved cases. The non--relaxable... |

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Citation Context ..., 28, 75] and phase transitions [68], turbulence [30, 45], water waves 3 [66], viscoelasticity 4 [60] and reactive flows 5 [9, 29, 50, 23]. Further, they govern models for the kinetic theory of gases =-=[17, 14, 7, 11, 33]-=-, and applications of these methods to kinetic--based schemes for compressible flow [27, 58, 56, 57] and rarefied--gas flows (including MHD and re--entry hypersonics) [25, 10, 13]. As is clear, such s... |

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Citation Context ...ation of the domain is detailed in Section 3.1. This is followed by the reconstruction stage described in Section 3.2, where our choice is the Piecewise Parabolic Method (PPM) of Colella and Woodward =-=[24]-=-. Recall that use of a higher-- order reconstruction does not in itself lead to higher accuracy for stiff problems (in the fluid dynamic limit) [38]. Rather, this choice of reconstruction was thought ... |

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Citation Context ...al phenomena, such as gas flows with relaxation 1 [37, 36, 8, 78, 20, 81, 82, 21, 65, 47, 48, 64, 42, 73, 5, 77, 12, 53, 52], multiphase flow 2 [32, 51, 28, 75] and phase transitions [68], turbulence =-=[30, 45]-=-, water waves 3 [66], viscoelasticity 4 [60] and reactive flows 5 [9, 29, 50, 23]. Further, they govern models for the kinetic theory of gases [17, 14, 7, 11, 33], and applications of these methods to... |

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Citation Context ...nd reactive flows 5 [9, 29, 50, 23]. Further, they govern models for the kinetic theory of gases [17, 14, 7, 11, 33], and applications of these methods to kinetic--based schemes for compressible flow =-=[27, 58, 56, 57]-=- and rarefied--gas flows (including MHD and re--entry hypersonics) [25, 10, 13]. As is clear, such systems form an extremely important class of problems, and are typically of the form w t + f x = \Gam... |

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Citation Context ..., 28, 75] and phase transitions [68], turbulence [30, 45], water waves 3 [66], viscoelasticity 4 [60] and reactive flows 5 [9, 29, 50, 23]. Further, they govern models for the kinetic theory of gases =-=[17, 14, 7, 11, 33]-=-, and applications of these methods to kinetic--based schemes for compressible flow [27, 58, 56, 57] and rarefied--gas flows (including MHD and re--entry hypersonics) [25, 10, 13]. As is clear, such s... |

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Citation Context ...s such properties is the point-- implicit method [83], where the source term is treated implicitly (locally). Studies focusing on numerical algorithm development for relaxation systems have been many =-=[31, 54, 55, 71]-=-, but only recently have good results been obtained for the difficult fluid dynamic limit by Jin and Levermore [41], Jin [38] and Caflisch, Jin and Russo [15]. It was shown there that poor results are... |

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Citation Context ...ilibrium. Such systems describe a wide variety of physical phenomena, such as gas flows with relaxation 1 [37, 36, 8, 78, 20, 81, 82, 21, 65, 47, 48, 64, 42, 73, 5, 77, 12, 53, 52], multiphase flow 2 =-=[32, 51, 28, 75]-=- and phase transitions [68], turbulence [30, 45], water waves 3 [66], viscoelasticity 4 [60] and reactive flows 5 [9, 29, 50, 23]. Further, they govern models for the kinetic theory of gases [17, 14, ... |

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Citation Context ...nd reactive flows 5 [9, 29, 50, 23]. Further, they govern models for the kinetic theory of gases [17, 14, 7, 11, 33], and applications of these methods to kinetic--based schemes for compressible flow =-=[27, 58, 56, 57]-=- and rarefied--gas flows (including MHD and re--entry hypersonics) [25, 10, 13]. As is clear, such systems form an extremely important class of problems, and are typically of the form w t + f x = \Gam... |

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Citation Context ...perbolic systems of conservation laws, where the source terms cause a "relaxation" towards equilibrium. Such systems describe a wide variety of physical phenomena, such as gas flows with rel=-=axation 1 [37, 36, 8, 78, 20, 81, 82, 21, 65, 47, 48, 64, 42, 73, 5, 77, 12, 53, 52]-=-, multiphase flow 2 [32, 51, 28, 75] and phase transitions [68], turbulence [30, 45], water waves 3 [66], viscoelasticity 4 [60] and reactive flows 5 [9, 29, 50, 23]. Further, they govern models for t... |

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Citation Context ...ead to the heat equation ` t = ` xx : However, this gives the paradoxical result of an infinite propagation speed for a diffusion problem. A convenient way around this is to assume a relaxation model =-=[16, 72, 47]-=-, the simplest version of which is q t + ` x = \Gammaq ; and wheresis a relaxation time. This leads to the Hyperbolic Heat Equations (HHE's), which are 2 ` t + q x = 0 ; (2.3) q t + 1 ` x = \Gamma q :... |

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Citation Context ...n this dissertation to be of uniform width \Deltax (Figure 3.1). However, the formulation can be extended quite readily to non--uniform meshes, such as those that arise in adaptive--grid computations =-=[6, 26, 59]-=-. In the finite--volume formulation, the cell averaged value in cell 41 j at time level t n (w n j ), defined by v v v j j \Gamma 1 j + 1 t n t n+1 t n+ff 3 t n+ff 2 t n+ff 1 v v v j \Gamma 1=2 j + 1=... |

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Citation Context ... = 0. The analytical solution is shown in Figure 2.2, where the exponential decay of the jumps along the characteristics is clearly observed. The solution for q in the region POQ (see Figure 2.3), is =-=[63, 74]-=- q(i; j) = / ` L \Gamma ` R 2 1 2 ! exp / \Gamma i + j 4 ! I 0 0 @ s ij 4 2 1 A = / ` L \Gamma ` R 2 1 2 ! exp ` \Gamma t 2 ' I 0 0 @ s t 2 \Gamma x 2 4 2 1 A ; (2.13) where I 0 is the modified Bessel... |

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Citation Context ...al (hence parallelizable) and yet not having the time step constrained by stability to be smaller than the relaxation time. A promising variant that has such properties is the point-- implicit method =-=[83]-=-, where the source term is treated implicitly (locally). Studies focusing on numerical algorithm development for relaxation systems have been many [31, 54, 55, 71], but only recently have good results... |

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Citation Context ...ty arises in the context of adaptive Cartesian--grid methods for the Navier--Stokes equations [22]. These methods are being increasingly used to solve complex flows (such as Euler [6, 26, 59] and MHD =-=[35]-=- flows, among others), and are thought to hold immense promise even for viscous flows (once these problem issues are resolved). Hence, they provide a strong motivation for this work. It was shown in [... |

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Citation Context ...al phenomena, such as gas flows with relaxation 1 [37, 36, 8, 78, 20, 81, 82, 21, 65, 47, 48, 64, 42, 73, 5, 77, 12, 53, 52], multiphase flow 2 [32, 51, 28, 75] and phase transitions [68], turbulence =-=[30, 45]-=-, water waves 3 [66], viscoelasticity 4 [60] and reactive flows 5 [9, 29, 50, 23]. Further, they govern models for the kinetic theory of gases [17, 14, 7, 11, 33], and applications of these methods to... |