Similarity Solutions of Shock Formation for First-Order Strictly Hyperbolic Systems

Shocks due to hyperbolic partial differential equations (PDEs) appear throughout mathematics and science. The canonical example is shock formation in the inviscid Burgers equation [Formula: see text]. Previous studies have shown that when shocks form for the inviscid Burgers equation, for positions and times close to the shock singularity, the dynamics are locally self-similar and universal, i.e., dynamics are equivalent regardless of the initial conditions. In this paper, we show that, in fact, shock formation is self-similar and universal for general first-order strictly hyperbolic PDEs in one spatial dimension, and the self-similarity is like that of the inviscid Burgers equation. An analytical formula is derived for the self-similar universal solution. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SIREV and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://doi.org/10.34770/6vgh-br25 and in the supplementary materials ( M180373_SM.zip [25.7MB]). [Formula: see text]