Abstract
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.
Highlights
The shallow water equations have a wide variety of applications in ocean and hydraulic engineering
A simple and efficient method for solving the equations with source terms is a split-step method 1, 2 in which the nonhomogeneous equations are splitt into a homogeneous equation and a set of ordinary differential equations dealing with only the source and time derivative terms; see Mathematical Problems in Engineering
The accuracy of numerical solutions are demonstrated in various test cases with and without friction effects, as well as the problems of wet and dry beds
Summary
The shallow water equations have a wide variety of applications in ocean and hydraulic engineering. The approximations are performed by the piecewise-linear reconstruction to calculate the conservative variables at cell interfaces in each cell volume These predicted solutions are employed to calculate the values of source terms at the corresponding half-time step. This is a different point from the usual operator splitting of the WAF method. The WAF method with the total variation diminishing TVD approach 7 is applied to obtain highorder solutions and to remove some oscillations in case of very high gradient problem.
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