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

• Published in: International journal for numerical methods in fluids 2021-01, Vol.93 (1), p.24-43
• Main Authors: Pitt, Jordan P.A., Zoppou, Christopher, Roberts, Stephen G.
• Format: Article
• Language: English
• Subjects: Bathymeters; Bathymetry; dry bed; Drying; Exact solutions; Finite element method; Finite volume method; hydrodynamics; incompressible flow; Lakes; Mathematical models; Nonlinear equations; Numerical analysis; Numerical methods; Robustness (mathematics); Serre equations; Solitary waves; Water depth; Wetting
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.4873

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. A new numerical method for solving the Serre equations for flows over dry beds is described. This numerical method is then validated against forced solutions, demonstrating its second‐order accuracy in the presence of dry beds.