A large time step upwind scheme for the shallow water equations with source terms

A modified predictor-corrector scheme combining with the depth gradient method (DGM) and the weighted average flux (WAF) method has been presented to solve the one-dimensional shallow water equations with source terms. Approximate solutions in the predictor step are obtained by the DGM with piecewise-linear reconstructions in each cell volume. The source terms can then be calculated directly by these predicted values at the corresponding half-time step. In the corrector step, the TVD version of the WAF method is applied to calculate the numerical fluxes at the same half-time step for each cell face. The accuracy of numerical solutions is shown by applying the method to solve various test cases in both steady and unsteady problems with and without source terms. It shows that the numerical results are in good agreement with the existing analytical solutions as well as experimental data in some test cases.

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

Communications in Computational Physics ◽

10.4208/cicp.040413.160114a ◽

2014 ◽

Vol 16 (2) ◽

pp. 307-347 ◽

Cited By ~ 17

Author(s):

Georgij Bispen ◽

K. R. Arun ◽

Mária Lukáčová-Medvid’ová ◽

Sebastian Noelle

Keyword(s):

Shallow Water ◽

Froude Number ◽

Finite Volume ◽

Large Time ◽

Time Step ◽

Water Flows ◽

Shallow Water Flows ◽

Nonlinear Advection ◽

Low Froude Number ◽

Large Time Step

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.

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Large Time Step Godunov Method for 2-D Conservation Laws

Acta Analysis Functionalis Applicata ◽

10.3724/sp.j.1160.2010.00097 ◽

2010 ◽

Vol 12 (2) ◽

pp. 97-109

Author(s):

Hairui WEN ◽

Jinghua WANG

Keyword(s):

Conservation Laws ◽

Large Time ◽

Godunov Method ◽

Time Step ◽

Large Time Step

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