2012/07/16 by Matthew C. Hernandez, Hernandez, Matthew
Mathematics · Engineering · #Advanced Mathematical Physics Problems #Computational Fluid Dynamics and Aerodynamics #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.1207.3857
We construct and justify leading order weakly nonlinear geometric optics\nexpansions for nonlinear hyperbolic initial value problems, including the\ncompressible Euler equations. The technique of simultaneous Picard iteration is\nemployed to show approximate solutions tend to the exact solutions in the small\nwavelength limit. Recent work [2] by Coulombel, Gues, and Williams studied the\ncase of reflecting wave trains whose expansions involve only real phases. We\ntreat generic boundary frequencies by incorporating into our expansions both\nreal and nonreal phases. Nonreal phases introduce difficulties such as\napproximately solving complex transport equations and result in the addition of\nboundary layers with exponential decay. This also prevents us from doing an\nerror analysis based on almost-periodic profiles as in [2].\n