A modified leapfrog scheme for shallow water equations

In this paper, we present an analytical and numerical study of staggered leapfrog scheme for linear shallow water equation. It is shown that the scheme is stable when Courant number < 1, has second order accurate in both time and space, and there is no damping error in this scheme. We implement the scheme to simulate standing wave in a closed basin to show that the surface motions stay zero in a node and have constant amplitude at the antinode. For an external force given into the basin, it will induce a resonance, which cause the wave amplitude is getting bigger at the position of antinode. Moreover, we simulate a wave in a tidal basin, and show that the model has infinite spin up time. For a linear shallow water equation with linear friction, it is shown that the model has finite spin up time.

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Solution of the linear shallow water equations by the fourth-order leapfrog scheme

Journal of Geophysical Research Atmospheres ◽

10.1029/93jc00652 ◽

1993 ◽

Vol 98 (C6) ◽

pp. 10205 ◽

Cited By ~ 3

Author(s):

Z. Kowalik

Keyword(s):

Shallow Water ◽

Fourth Order ◽

Shallow Water Equations ◽

Leapfrog Scheme

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Instability in Leapfrog and Forward–Backward Schemes

Monthly Weather Review ◽

10.1175/2009mwr3127.1 ◽

2010 ◽

Vol 138 (5) ◽

pp. 1497-1501 ◽

Cited By ~ 8

Author(s):

Wen-Yih Sun

Keyword(s):

Shallow Water ◽

Shallow Water Equations ◽

Numerical Schemes ◽

Courant Number ◽

Diffusion Term ◽

Repeated Eigenvalues ◽

Leapfrog Scheme

Abstract This paper shows that in the linearized shallow-water equations, the numerical schemes can become weakly unstable for the 2Δx wave in the C grid when the Courant number is 1 in the forward–backward scheme and 0.5 in the leapfrog scheme because of the repeated eigenvalues in the matrices. The instability can be amplified and spread to other waves and smaller Courant number if the diffusion term is included. However, Shuman smoothing can control the instability.

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MAPPING TIDAL CURRENTS AND RESIDUAL CURRENTS BY USE OF COASTAL ACOUSTIC TOMOGRAPHY

Proceedings of International Conference "Managinag risks to coastal regions and communities in a changinag world" (EMECS'11 - SeaCoasts XXVI) ◽

10.31519/conferencearticle_5b1b94407213d3.12732916 ◽

2017 ◽

Author(s):

Xiao-Hua Zhu ◽

Ze-Nan Zhu ◽

Xinyu Guo ◽

...

Keyword(s):

Shallow Water ◽

Shallow Water Equations ◽

Mean Amplitude ◽

Momentum Equation ◽

Tidal Currents ◽

Acoustic Tomography ◽

Mean Values ◽

Residual Currents ◽

Water Depths ◽

Deep Area

A coastal acoustic tomography (CAT) experiment for mapping the tidal currents in the Zhitouyang Bay was successfully carried out with seven acoustic stations during July 12 to 13, 2009. The horizontal distributions of tidal current in the tomography domain are calculated by the inverse analysis in which the travel time differences for sound traveling reciprocally are used as data. Spatial mean amplitude ratios M2 : M4 : M6 are 1.00 : 0.15 : 0.11. The shallow-water equations are used to analyze the generation mechanisms of M4 and M6. In the deep area, velocity amplitudes of M4 measured by CAT agree well with those of M4 predicted by the advection terms in the shallow water equations, indicating that M4 in the deep area where water depths are larger than 60 m is predominantly generated by the advection terms. M6 measured by CAT and M6 predicted by the nonlinear quadratic bottom friction terms agree well in the area where water depths are less than 20 m, indicating that friction mechanisms are predominant for generating M6 in the shallow area. Dynamic analysis of the residual currents using the tidally averaged momentum equation shows that spatial mean values of the horizontal pressure gradient due to residual sea level and of the advection of residual currents together contribute about 75% of the spatial mean values of the advection by the tidal currents, indicating that residual currents in this bay are induced mainly by the nonlinear effects of tidal currents.

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