Numerical solution of the fully non-linear weakly dispersive serre equations for steep gradient flows

Zoppou, Christopher; Pitt, Jordan; Roberts, Stephen

doi:10.1016/j.apm.2017.03.059

Cited by 9 publications

(31 citation statements)

References 43 publications

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“…The V i methods have demonstrated the appropriate order of convergence for smooth problems [4]. Furthermore, V 2 and V 3 have been validated against experimental data containing steep gradients [4]. The two methods D and E were found to be stable under the same CFL condition.…”

Section: Methodsmentioning

confidence: 94%

“…Five numerical schemes were used to investigate the behaviour of the Serre equations in the presence of steep gradients, the first (V 1 ), second (V 2 ) and third-order (V 3 ) finite difference finite volume methods of Zoppou et al [4], the second-order finite difference method of El et al [7] (E) and a second-order finite difference method (D) that can be found in the Appendix.…”

Section: Methodsmentioning

confidence: 99%

“…The Serre equations are used as a compromise between the non-dispersive shallow water wave equations and the incompressible inviscid Euler equations for modelling dispersive waves of the free surface in the presence of steep gradients, which are present for the Euler equations [1] but not for the shallow water wave equations. The Serre equations like the shallow water wave equations produce methods [2][3][4] that are computationally easier and quicker to solve than the best methods for the Euler equations. The Serre equations are considered the most appropriate approximation to the Euler equations for modelling dispersive waves up to the shore line [5,6].…”

Section: Introductionmentioning

confidence: 99%

“…There are few examples in the literature which depict the behaviour of numerical solutions to the Serre equations in the presence of steep gradients [1-4, 7, 8]. These papers all present problems with discontinuous initial conditions [2][3][4] or a smooth approximation to them when the numerical method requires some smoothness of the solutions [1,7,8]. Among these papers there are differences in the structures of the numerical solutions, with some demonstrating undulations in depth and velocity throughout the bore [3,4,7] and others showing a constant depth and velocity state in the middle of the bore [1,2,8].…”

Section: Introductionmentioning

confidence: 99%

“…The numerical solution is also consistent across the five different numerical methods. Three of the methods are the first, second and third-order methods presented by Zoppou et al [4]. The first-order method is equivalent to the method of Le Métayer et al [2].…”

Section: Introductionmentioning

confidence: 99%

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Behaviour of the Serre equations in the presence of steep gradients revisited

2018

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We use numerical methods to study the behaviour of the Serre equations in the presence of steep gradients because there are no known analytical solutions for these problems. In keeping with the literature we study a class of initial condition problems that are a smooth approximation to the initial conditions of the dam-break problem. This class of initial condition problems allow us to observe the behaviour of the Serre equations with varying steepness of the initial conditions. The numerical solutions of the Serre equations are justified by demonstrating that as the resolution increases they converge to a solution with little error in conservation of mass, momentum and energy independent of the numerical method. We observe four different structures of the converged numerical solutions depending on the steepness of the initial conditions. Two of these structures were observed in the literature, with the other two not being commonly found in the literature. The numerical solutions are then used to assess how well the analytical solution of the shallow water wave equations captures the mean behaviour of the solution of the Serre equations for the dam-break problem. Lastly the numerical solutions are used to evaluate the usefulness of asymptotic results in the literature to approximate the depth and location of the front of an undular bore.

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Section: Methodsmentioning

confidence: 94%

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Behaviour of the Serre equations in the presence of steep gradients revisited

2018

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Solving the fully nonlinear weakly dispersive Serre equations for flows over dry beds

Pitt

Zoppou

Roberts

2020

Numerical Methods in Fluids

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We describe a numerical method for solving the Serre equations that can simulate flows over dry bathymetry. The method solves the Serre equations in conservation law form with a finite volume method. A finite element method is used to solve the auxiliary elliptic equation for the depth-averaged horizontal velocity. The numerical method is validated against the lake at rest analytic solution, demonstrating that it is well-balanced. Since there are currently no known nonstationary analytical solutions to the Serre equation that involve bathymetry, a nonstationary forced solution, involving bathymetry was developed. The method was further validated and its convergence rate established using the developed nonstationary forced solution containing the wetting and drying of bathymetry. Finally, the method is also validated against experimental results for the run-up of a solitary wave on a sloped beach. The finite-volume finite-element approach to solving the Serre equation was found to be accurate and robust.

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