Communications in Mathematical Sciences

Volume 18 (2020)

Number 4

Effective Rankine–Hugoniot conditions for shock waves in periodic media

Pages: 1023 – 1040

DOI: https://dx.doi.org/10.4310/CMS.2020.v18.n4.a6

Authors

David I. Ketcheson (Computer, Electrical, & Mathematical Sciences & Engineering Division, King Abdullah University of Science & Technology (KAUST), Thuwal, Saudi Arabia)

Manuel Quezada de Luna (Computer, Electrical, & Mathematical Sciences & Engineering Division, King Abdullah University of Science & Technology (KAUST), Thuwal, Saudi Arabia)

Abstract

Solutions of first-order nonlinear hyperbolic conservation laws typically develop shocks in finite 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.

Keywords

shock wave, periodic medium, dispersion, homogenization

2010 Mathematics Subject Classification

35B27, 35L60, 35L67

This work was supported by funding from King Abdullah University of Science & Technology (KAUST).

Received 11 September 2019

Accepted 13 January 2020