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

AbstractWe 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.

Download Full-text

Assessing the Numerical Accuracy of Complex Spherical Shallow-Water Flows

Monthly Weather Review ◽

10.1175/2007mwr2036.1 ◽

2007 ◽

Vol 135 (11) ◽

pp. 3876-3894 ◽

Cited By ~ 14

Author(s):

Ali R. Mohebalhojeh ◽

David G. Dritschel

Keyword(s):

Shallow Water ◽

Fourth Order ◽

Large Scale ◽

Numerical Accuracy ◽

Conservation Of Energy ◽

Water Flows ◽

Shallow Water Flows ◽

Low Froude Number ◽

Contour Advection ◽

High Reynolds

Abstract The representation of nonlinear shallow-water flows poses severe challenges for numerical modeling. The use of contour advection with contour surgery for potential vorticity (PV) within the contour-advective semi-Lagrangian (CASL) algorithm makes it possible to handle near-discontinuous distributions of PV with an accuracy beyond what is accessible to conventional algorithms used in numerical weather and climate prediction. The emergence of complex distributions of the materially conserved quantity PV, in the absence of forcing and dissipation, results from large-scale shearing and deformation and is a common feature of high Reynolds number flows in the atmosphere and oceans away from boundary layers. The near-discontinuous PV in CASL sets a limit on the actual numerical accuracy of the Eulerian, grid-based part of CASL. For the spherical shallow-water equations, the limit is studied by comparing the accuracy of CASL algorithms with second-order-centered, fourth-order-compact, and sixth-order-supercompact finite differencing in latitude in conjunction with a spectral treatment in longitude. The comparison is carried out on an unstable midlatitude jet at order one Rossby number and low Froude number that evolves into complex vortical structures with sharp gradients of PV. Quantitative measures of global conservation of energy and angular momentum, and of imbalance as diagnosed using PV inversion by means of Bolin–Charney balance, indicate that fourth-order differencing attains the highest numerical accuracy achievable for such nonlinear, advectively dominated flows.

Download Full-text

A staggered conservative scheme for every Froude number in rapidly varied shallow water flows

International Journal for Numerical Methods in Fluids ◽

10.1002/fld.537 ◽

2003 ◽

Vol 43 (12) ◽

pp. 1329-1354 ◽

Cited By ~ 240

Author(s):

G. S. Stelling ◽

S. P. A. Duinmeijer

Keyword(s):

Shallow Water ◽

Froude Number ◽

Conservative Scheme ◽

Water Flows ◽

Shallow Water Flows

Download Full-text

The classical hydraulic jump in a model of shear shallow-water flows

Journal of Fluid Mechanics ◽

10.1017/jfm.2013.174 ◽

2013 ◽

Vol 725 ◽

pp. 492-521 ◽

Cited By ~ 54

Author(s):

G. L. Richard ◽

S. L. Gavrilyuk

Keyword(s):

Shallow Water ◽

Froude Number ◽

Hydraulic Jump ◽

Surface Profile ◽

Stationary Solutions ◽

Water Flows ◽

Shallow Water Flows ◽

Free Surface Profile ◽

Sequent Depth ◽

Two Parameter

AbstractA conservative hyperbolic two-parameter model of shear shallow-water flows is used to study the classical turbulent hydraulic jump. The parameters of the model, which are the wall enstrophy and the roller dissipation coefficient, are determined from measurements of the roller length and the deviation from the Bélanger equation of the sequent depth ratio (experimental data by Hager & Bremen, J. Hydraul. Res., vol. 27, 1989, pp. 565–585; and Hager, Bremen & Kawagoshi, J. Hydraul. Res., vol. 28, 1990, pp. 591–608). Stationary solutions to the model describe with a good accuracy the free-surface profile of the hydraulic jump. The model is also capable of predicting the oscillations of the jump toe. We show that if the upstream Froude number is larger than ${\sim }1. 5$, the jump toe oscillates with a particular frequency, while for the Froude number smaller than 1.5 the solution becomes stationary. In particular, we show that for a given flow discharge, the oscillation frequency is a decreasing function of the Froude number.

Download Full-text