Vater, Stefan
[Author]
;
Rupert Klein
[Contributor];
Michael L. Minion
[Contributor]
A multigrid-based multiscale numerical scheme for shallow water flows at low froude number ; Ein mehrgitterbasiertes Mehrskalenverfahren für Flachwasserströmungen bei kleiner Froudezahl
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Media type:
Doctoral Thesis;
Electronic Thesis;
E-Book
Title:
A multigrid-based multiscale numerical scheme for shallow water flows at low froude number ; Ein mehrgitterbasiertes Mehrskalenverfahren für Flachwasserströmungen bei kleiner Froudezahl
Contributor:
Vater, Stefan
[Author]
Published:
Freie Universität Berlin: Refubium (FU Berlin), 2013
Footnote:
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Description:
Abstract 1 Introduction 2 Governing equations 3 A multilevel method for long- wave linear shallow water flows 4 Numerical solution of the "lake equations" 5 Semi-implicit solution of low Froude number shallow water flows 6 Discussion Appendix Bibliography ; A new multiscale semi-implicit scheme for the computation of low Froude number shallow water flows is presented. Motivated by the needs of atmospheric flow applications, it aims to minimize dispersion and amplitude errors in the computation of long-wave gravity waves. While it correctly balances "slaved" dynamics of short-wave solution components induced by slow forcing, the method eliminates freely propagating compressible short-wave modes, which are under- resolved in time. This is achieved through a multilevel approach borrowing ideas from multigrid schemes for elliptic equations. The scheme is second- order accurate and admits time steps depending essentially on the flow velocity. First, a multilevel method is derived for the one-dimensional linearized shallow water equations. Scale-wise decomposition of the data enables a scale-dependent blending of time integrators with different principal features. To guide the selection of these integrators, the discrete- dispersion relations of some standard second-order schemes are analyzed, and their response to high-wave-number low-frequency source terms is discussed. The resulting method essentially consists of the solution of a Helmholtz problem on the original fine grid, where the differencing operator and the right hand side incorporate the multiscale information of the discretization. The performance of the new multilevel method is illustrated on a test case with "multiscale" initial data and a problem with a slowly varying high-wave- number source term. The scheme for simulating fully nonlinear shallow water flows is a generalization of a projection method for the zero Froude number equations. Therefore, the method described in Vater and Klein (Numer. Math. 113, pp. 123-161, 2009) is extended to account for ...