## Convergence of relaxation schemes to the equations of elastodynamics (2000)

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Venue: | MATH. COMP |

Citations: | 7 - 1 self |

### BibTeX

@ARTICLE{Gosse00convergenceof,

author = {Laurent Gosse and Athanasios E. Tzavaras},

title = { Convergence of relaxation schemes to the equations of elastodynamics},

journal = {MATH. COMP},

year = {2000},

volume = {70},

pages = {70--555}

}

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### Abstract

We study the effect of approximation matrices to semi-discrete relaxation schemes for the equations of one-dimensional elastodynamics. We consider a semi-discrete relaxation scheme and establish convergence using the L p theory of compensated compactness. Then we study the convergence of an associated relaxation-diffusion system, inspired by the scheme. Numerical comparisons of fully-discrete schemes are carried out.

### Citations

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Citation Context ...1 εE (σ − g(u)), vt − σx = havxx, (σ − Eu)t = − 1 (34) ε (σ − g(u)). Due to (6), the reaction term in (34) is globally Lipshitz and global existence follows from general semigroup theory (e.g., Henry =-=[8]-=-). If the data v0, σ0 ∈ H 2 (R), u0 ∈ H 1 (R), then for any T>0 there exists a unique globally defined solution (35) v, σ ∈ C([0,T],H 2 (R)) , u ∈ C([0,T],H 1 (R)) . We assume that the data satisfy th... |

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Citation Context ...,σj(0)) in ℓ∞ (Z), there exists a unique solution of (25), (uj,vj,σj) ∈ C 1 ([0, +∞),ℓ ∞ (Z)) .562 LAURENT GOSSE AND ATHANASIOS E. TZAVARAS This is a consequence of the Cauchy-Lipschitz theorem (cf. =-=[3]-=- p. 104), since each equation in (25) depends on a finite number of values. We define for each h>0,ε>0 approximations of (2) denoted by such that (27) (u h,ε ,v h,ε ,σ h,ε ) ∈ C 1 ([0, +∞),L ∞ (R)) (u... |

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Citation Context ...able, the upwind discretization of (2) is far less diffusive than the upwind discretization of (4). The main ingredients for both convergence results are the theory of compensated compactness (Tartar =-=[20]-=-, Murat [14]), the L p theory for the reduction of generalized Young measures for the equations of elastodynamics (DiPerna [6], Shearer [19], Serre-Shearer [18]), and a priori estimates—valid under th... |

157 | The relaxation schemes for system of conservation laws in arbitrary space dimensions
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Citation Context ...ation laws appear in diverse models in continuum mechanics and kinetic theory of gases, and serve as a groundstage for the design of numerical schemes for hyperbolic systems of conservation laws (see =-=[9, 4, 5, 23, 2]-=- for a range of perspectives and [16] for the related subject of kinetic schemes). The convergence properties of relaxation systems and associated relaxation schemes for scalar conservation laws are p... |

123 | Hyperbolic conservation laws with stiff relaxation terms and entropy
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Citation Context ...ation laws appear in diverse models in continuum mechanics and kinetic theory of gases, and serve as a groundstage for the design of numerical schemes for hyperbolic systems of conservation laws (see =-=[9, 4, 5, 23, 2]-=- for a range of perspectives and [16] for the related subject of kinetic schemes). The convergence properties of relaxation systems and associated relaxation schemes for scalar conservation laws are p... |

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Citation Context ...th convergence results are the theory of compensated compactness (Tartar [20], Murat [14]), the L p theory for the reduction of generalized Young measures for the equations of elastodynamics (DiPerna =-=[6]-=-, Shearer [19], Serre-Shearer [18]), and a priori estimates—valid under the hypothesis 0 <g ′ <E— measuring the dissipative strength of the semi-discrete relaxation scheme (9) and of the relaxation-di... |

70 | Convergence to equilibrium for the relaxation approximation of conservation laws
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Citation Context ... [16] for the related subject of kinetic schemes). The convergence properties of relaxation systems and associated relaxation schemes for scalar conservation laws are presently well understood (e.g., =-=[4, 15, 1, 21, 10, 11]-=-). By contrast, when the zero-relaxation limit is a system of conservation laws, the dissipative effect of relaxation is subtle to capture and convergence results were only recently established [22, 2... |

38 |
Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics
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Citation Context ...ation laws appear in diverse models in continuum mechanics and kinetic theory of gases, and serve as a groundstage for the design of numerical schemes for hyperbolic systems of conservation laws (see =-=[9, 4, 5, 23, 2]-=- for a range of perspectives and [16] for the related subject of kinetic schemes). The convergence properties of relaxation systems and associated relaxation schemes for scalar conservation laws are p... |

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Citation Context ... [16] for the related subject of kinetic schemes). The convergence properties of relaxation systems and associated relaxation schemes for scalar conservation laws are presently well understood (e.g., =-=[4, 15, 1, 21, 10, 11]-=-). By contrast, when the zero-relaxation limit is a system of conservation laws, the dissipative effect of relaxation is subtle to capture and convergence results were only recently established [22, 2... |

28 |
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Citation Context ...0, 11]). By contrast, when the zero-relaxation limit is a system of conservation laws, the dissipative effect of relaxation is subtle to capture and convergence results were only recently established =-=[22, 23, 17, 12]-=-. The issue of dependence on the approximation matrix, familiar from the theory of viscosity approximations, has a counterpart in the theory of relaxation approximations and their associated relaxatio... |

25 | On the rate of convergence to equilibrium for a system of conservation laws with a relaxation term
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Citation Context ... [16] for the related subject of kinetic schemes). The convergence properties of relaxation systems and associated relaxation schemes for scalar conservation laws are presently well understood (e.g., =-=[4, 15, 1, 21, 10, 11]-=-). By contrast, when the zero-relaxation limit is a system of conservation laws, the dissipative effect of relaxation is subtle to capture and convergence results were only recently established [22, 2... |

22 | Materials with internal variables and relaxation to conservation
- Tzavaras
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Citation Context |

21 | L’injection du cône positif de H −1 dans W −1,q est compacte pour tout q<2 - Murat - 1981 |

10 |
Convergence of a relaxation scheme for hyperbolic systems of conservation laws. Numer. Math., vol 88, pp 121-134 (2001). Benjamin Boutin UMR 7598 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie-Paris6, Boîte courrier 187, 75252 PARIS CED
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Citation Context ...0, 11]). By contrast, when the zero-relaxation limit is a system of conservation laws, the dissipative effect of relaxation is subtle to capture and convergence results were only recently established =-=[22, 23, 17, 12]-=-. The issue of dependence on the approximation matrix, familiar from the theory of viscosity approximations, has a counterpart in the theory of relaxation approximations and their associated relaxatio... |

9 | Global existence and compactness in L p for the quasi-linear wave equation - Shearer - 1994 |

8 |
Convergence and error estimates of relaxation schemes for multidimensional conservation laws
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Citation Context |

7 |
Convergence with Physical Viscosity for Nonlinear Elasticity, manuscript
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Citation Context ...eory of compensated compactness (Tartar [20], Murat [14]), the L p theory for the reduction of generalized Young measures for the equations of elastodynamics (DiPerna [6], Shearer [19], Serre-Shearer =-=[18]-=-), and a priori estimates—valid under the hypothesis 0 <g ′ <E— measuring the dissipative strength of the semi-discrete relaxation scheme (9) and of the relaxation-diffusion system (10). The a priori ... |

5 |
The energy in one-dimensional rate-type semilinear viscoelasticity
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Citation Context ...es support of the National Science Foundation and the Office for Naval Research. 555 c○2000 American Mathematical Society556 LAURENT GOSSE AND ATHANASIOS E. TZAVARAS It is a model in viscoelasticity =-=[7, 23]-=-, and it may be put into the equivalent form ut − vx = 0, vt − g(u)x = ε ( (3) ) Evxx − vtt , of an approximation of (1) by one wave equation. The second system is of the type proposed by Jin and Xin ... |

5 |
Viscosity and relaxation approximations for hyperbolic systems of conservation laws. In An Introduction to Recent Developments in Theory and Numerics for Conservation
- Tzavaras
- 1998
(Show Context)
Citation Context ...0, 11]). By contrast, when the zero-relaxation limit is a system of conservation laws, the dissipative effect of relaxation is subtle to capture and convergence results were only recently established =-=[22, 23, 17, 12]-=-. The issue of dependence on the approximation matrix, familiar from the theory of viscosity approximations, has a counterpart in the theory of relaxation approximations and their associated relaxatio... |

4 |
Cauchy problem for hyperbolic conservation laws with a relaxation term
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Citation Context ... a more stable response. Indeed, this is the case for a relaxationdiffusion system that looks exactly like (10) with the notable exception that the term σxx is replaced by uxx (see Lu and Klingenberg =-=[13]-=-). However, for the system (10) this is not the case; the slight difference in terms makes a large difference in the analysis, which is based on using the stabilizing effect of relaxation to compensat... |

3 |
An introduction to kinetic schemes for gas dynamics. An introduction to recent developments in theory and numerics for conservation laws
- Perthame
- 1998
(Show Context)
Citation Context ...mechanics and kinetic theory of gases, and serve as a groundstage for the design of numerical schemes for hyperbolic systems of conservation laws (see [9, 4, 5, 23, 2] for a range of perspectives and =-=[16]-=- for the related subject of kinetic schemes). The convergence properties of relaxation systems and associated relaxation schemes for scalar conservation laws are presently well understood (e.g., [4, 1... |

2 |
L’injection du cône positif de H−1 dans W −1,q est compacte pour tout q<2
- Murat
- 1981
(Show Context)
Citation Context ...wind discretization of (2) is far less diffusive than the upwind discretization of (4). The main ingredients for both convergence results are the theory of compensated compactness (Tartar [20], Murat =-=[14]-=-), the L p theory for the reduction of generalized Young measures for the equations of elastodynamics (DiPerna [6], Shearer [19], Serre-Shearer [18]), and a priori estimates—valid under the hypothesis... |

2 |
Global existence and compactness in Lp for the quasi-linear wave equation
- Shearer
- 1994
(Show Context)
Citation Context ...e results are the theory of compensated compactness (Tartar [20], Murat [14]), the L p theory for the reduction of generalized Young measures for the equations of elastodynamics (DiPerna [6], Shearer =-=[19]-=-, Serre-Shearer [18]), and a priori estimates—valid under the hypothesis 0 <g ′ <E— measuring the dissipative strength of the semi-discrete relaxation scheme (9) and of the relaxation-diffusion system... |