G Stoltz 2005 Nonlinearity 18 1967 doi:10.1088/0951-7715/18/5/006
G Stoltz
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We derive here a simplified discrete one-dimensional model describing some important features of shock waves. In order to avoid expensive multidimensional simulations, one-dimensional models are commonly used, but the existing ones often exhibit some spurious physically irrelevant behaviour. Here we build a one-dimensional model with perturbations arising from mean higher-dimensional behaviour. The coupling of the system with a deterministic heat bath in the Kac–Zwanzig fashion allows us to derive a generalized Langevin equation for the system, without a priori fixing the temperature in the shocked region. This deterministic problem with several degrees of freedom is then reduced to a simpler stochastic problem with memory. Some numerical results are provided, which illustrate and confirm the qualitative correctness of the model.
62.50.-p High-pressure effects in solids and liquids
05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin, etc.)
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) (See also 60H10)
Issue 5 (September 2005)
Received 17 November 2004, in final form 5 May 2005
Published 10 June 2005
G Stoltz 2005 Nonlinearity 18 1967
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