Vol. 13, No. 2, 2018

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A semi-implicit multiscale scheme for shallow water flows at low Froude number

Stefan Vater and Rupert Klein

Vol. 13 (2018), No. 2, 303–336
Abstract

A new large time step semi-implicit multiscale method is presented for the solution of low Froude number shallow water flows. While on small scales which are under-resolved in time the impact of source terms on the divergence of the flow is essentially balanced, on large resolved scales the scheme propagates free gravity waves with minimized diffusion. The scheme features a scale decomposition based on multigrid ideas. Two different time integrators are blended at each scale depending on the scale-dependent Courant number for gravity wave propagation. The finite volume discretization is implemented in the framework of second-order Godunov-type methods for conservation laws. The basic properties of the method are validated by numerical tests. This development is a further step in the construction of asymptotically adaptive numerical methods for the computation of large-scale atmospheric flows.

Keywords
shallow water equations, multiscale time integration, asymptotically adaptive numerical methods, large time steps, balanced modes
Mathematical Subject Classification 2010
Primary: 65M08, 86A10
Milestones
Received: 1 December 2017
Revised: 27 June 2018
Accepted: 16 July 2018
Published: 25 September 2018
Authors
Stefan Vater
Department of Mathematics and Computer Science
Freie Universität Berlin
Berlin
Germany
Rupert Klein
Department of Mathematics and Computer Science
Freie Universität Berlin
Berlin
Germany