Effective rankine-hugoniot conditions for shock waves in periodic media
KAUST DepartmentApplied Mathematics and Computational Science Program
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Numerical Mathematics Group
Preprint Posting Date2019-09-11
Online Publication Date2020-07-28
Print Publication Date2020
Embargo End Date2021-01-13
Permanent link to this recordhttp://hdl.handle.net/10754/659990
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AbstractSolutions of first-order nonlinear hyperbolic conservation laws typically develop shocks infinite time even from smooth initial conditions. However, in heterogeneous media with rapid spatial variation, shock formation may be delayed or avoided. When shocks do form in such media, their speed of propagation depends on the material structure. We investigate conditions for shock formation and propagation in heterogeneous media. We focus on the propagation of plane waves in two-dimensional media with a periodic structure that changes in only one direction. We propose an estimate for the speed of the shocks that is based on the Rankine-Hugoniot conditions applied to a leading-order homogenized (constant coefficient) system. We verify this estimate via numerical simulations using different nonlinear constitutive relations and layered and smoothly varying media with a periodic structure. In addition, we discuss conditions and regimes under which shocks form in this type of media.
CitationKetcheson, D. I., & Quezada de Luna, M. (2020). Effective Rankine–Hugoniot conditions for shock waves in periodic media. Communications in Mathematical Sciences, 18(4), 1023–1040. doi:10.4310/cms.2020.v18.n4.a6
SponsorsThis work was supported by funding from King Abdullah University of Science & Technology (KAUST).
PublisherInternational Press of Boston