2023/09/16 by Yang, Zhao, Kevin Zumbrun, Zumbrun, Kevin
Engineering · Environmental Science · #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Vibration Analysis #Hydraulic flow and structures #Hydrology and Sediment Transport Processes
paper · pdf · doi:10.48550/arxiv.2309.08870
openalex publication_date 2023/09/16 · openalex created_date 2023/09/20 · openalex updated_date 2026/07/28
We study by a combination of analytical and numerical methods multidimensional stability and transverse bifurcation of planar hydraulic shock and roll wave solutions of the inviscid Saint Venant equations for inclined shallow-water flow, both in the whole space and in a channel of finite width, obtaining complete stability diagrams across the full parameter range of existence. Technical advances include development of efficient multi-d Evans solvers, low- and high-frequency asymptotics, explicit/semi-explicit computation of stability boundaries, and rigorous treatment of channel flow with wall-type physical boundary. Notable behavioral phenomena are a novel essential transverse bifurcation of hydraulic shocks to invading planar periodic roll-wave or doubly-transverse periodic herringbone patterns, with associated metastable behavior driven by mixed roll- and herringbone-type waves initiating from localized perturbation of an unstable constant state; and Floquet-type transverse ``flapping'' bifurcation of roll wave patterns.