Commun. Comput. Phys., 16 (2014), pp. 307-347.


IMEX Large Time Step Finite Volume Methods for Low Froude Number Shallow Water Flows

Georgij Bispen 1, K. R. Arun 2, Maria Lukacova-Medvid'ova 1*, Sebastian Noelle 3

1 Institute of Mathematics, University of Mainz, Germany.
2 School of Mathematics, Indian Institute of Science Education and Research Thiruvananthapuram, India.
3 IGPM, RWTH Aachen, Germany.

Received 4 April 2013; Accepted (in revised version) 16 January 2014
Available online 17 April 2014
doi:10.4208/cicp.040413.160114a

Abstract

We present new large time step methods for the shallow water flows in the low Froude number limit. In order to take into account multiscale phenomena that typically appear in geophysical flows nonlinear fluxes are split into a linear part governing the gravitational waves and the nonlinear advection. We propose to approximate fast linear waves implicitly in time and in space by means of a genuinely multidimensional evolution operator. On the other hand, we approximate nonlinear advection part explicitly in time and in space by means of the method of characteristics or some standard numerical flux function. Time integration is realized by the implicit-explicit (IMEX) method. We apply the IMEX Euler scheme, two step Runge Kutta Cranck Nicolson scheme, as well as the semi-implicit BDF scheme and prove their asymptotic preserving property in the low Froude number limit. Numerical experiments demonstrate stability, accuracy and robustness of these new large time step finite volume schemes with respect to small Froude number.

AMS subject classifications: 35L65, 76B15, 65M08, 65M06, 35L45, 35L65

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Key words: Low Froude number flows, asymptotic preserving schemes, shallow water equations, large time step, semi-implicit approximation, evolution Galerkin schemes.

*Corresponding author.
Email: bispeng@uni-mainz.de (G. Bispen), arun@iisertvm.ac.in (K. R. Arun), lukacova@uni-mainz.de (M. Lukacova-Medvid'ova), noelle@igpm.rwth-aachen.de (S. Noelle)
 

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