2015/05/22 by Jean-François Coulombel, Coulombel, Jean-François · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.1505.06060
openalex publication_date 2015/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Leray and G aarding have developed a multiplier technique for deriving a\npriori estimates for solutions to scalar hyperbolic equations in either the\nwhole space or the torus. In particular, the arguments in Leray and\nG aarding's work provide with at least one local multiplier and one local\nenergy functional that is controlled along the evolution. The existence of such\na local multiplier is the starting point of the argument by Rauch for the\nderivation of semigroup estimates for hyperbolic initial boundary value\nproblems. In this article, we explain how this multiplier technique can be\nadapted to the framework of finite difference approximations of transport\nequations. The technique applies to numerical schemes with arbitrarily many\ntime levels, and encompasses a somehow magical trick that has been known for a\nlong time for the leapfrog scheme. More importantly, the existence and\nproperties of the local multiplier enable us to derive optimal semigroup\nestimates for fully discrete hyperbolic initial boundary value problems, which\nanswers a problem raised by Trefethen, Kreiss and Wu.\n