A new run-up algorithm based on local high-order analytic expansions

Gayaz Khakimzyanov (1) , Nina Shokina (1) , Denys Dutykh (2, 3, 4) , Dimitrios Mitsotakis (5)

1 Institute of Computational Technologies
2 LAMA - Laboratoire de Mathématiques
3 INSMI-CNRS - Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques
4 USMB [Université de Savoie] [Université de Chambéry] - Université Savoie Mont Blanc
5 Victoria University of Wellington

The practically important problem of the wave run-up is studied in this article in the framework of Nonlinear Shallow Water Equations (NSWE). The main novelty consists in the usage of high order local asymptotic analytical solutions in the vicinity of the shoreline. Namely, we use the analytical techniques introduced by S. Kovalevskaya and the analogy with the compressible gas dynamics (i.e. gas outflow problem into the vacuum). Our run-up algorithm covers all the possible cases of the wave slope on the shoreline and it incorporates the new analytical information in order to determine the shoreline motion to higher accuracy. The application of this algorithm is illustrated on several important practical examples. Finally, the simulation results are compared with the well-known analytical and experimental predictions.

Mots clés: wave run-up; asymptotic expansion; finite volumes; onlinear shallow water equations; finite differences

Domaines: Mécanique des fluides [physics.class-ph]; Analyse numérique [cs.NA]; Analyse classique [math.CA]; Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]; Dynamique des Fluides [physics.flu-dyn]; Physique Atmosphérique et Océanique [physics.ao-ph]; Physique Numérique [physics.comp-ph]

Gayaz Khakimzyanov, Nina Shokina, Denys Dutykh, Dimitrios Mitsotakis. A new run-up algorithm based on local high-order analytic expansions. Journal of Computational and Applied Mathematics, 2016, 298, pp.82-96. ⟨10.1016/j.cam.2015.12.004⟩. ⟨hal-01084811v3⟩