Development of an efficient wetting and drying treatment for shallow‐water modeling using the quadrature‐free Runge‐Kutta discontinuous Galerkin method

Abstract

SummaryTo develop a robust, well‐balanced and quadrature‐free Runge‐Kutta discontinuous Galerkin (RKDG) shallow water solver, we introduce an efficient wetting and drying (WD) treatment in this paper. The main feature of this WD treatment is the use of vertex‐based linear reconstructed solutions in transition (partially wet) regions and high‐order solutions in smooth wet areas. To preserve the positivity of water depth, we also propose a modified time step size with the quadrature‐free scheme. The advantages of the WD treatment include robustness and the capability to address arbitrary high‐order RKDG methods in wet regions, thus making the quadrature‐free scheme more accurate and applicable to various flooding and drying problems. Two numerical test cases are used to validate the numerical method, and the results indicate that the WD method is accurate and robust for different flow regimes with dry areas.

A Well-Balanced Runge-Kutta Discontinuous Galerkin Method for Multilayer Shallow Water Equations with Non-Flat Bottom Topography

“…Finally, the method presented in this paper does not straightforwardly resolve we/dry areas in the multilayer shallow water flows. However, using ideas of wet/dry treatment developed in [30] for standard single-layer shallow water equations, it is possible to reconstruct an efficient wetting and drying treatment for two-dimensional multilayer shallow-water models using the Runge-Kutta discontinuous Galerkin method. Results on these techniques will be reported in future works.…”

“…Note that a similar hydrostatic reconstruction to (30) has been implemented in [5] for single-layer shallow water equations. The intermediate solutions…”

“…are obtained using the approximations (30). Once these numerical fluxes are reconstructed, the right hand side term in ( 27) is computed as a volume contribution (evaluated using the projected bed elevation Z h ) and a surface contribution of the previous fluxes.…”

A Wetting and Drying Approach for a Mode-Nonsplit Discontinuous Galerkin Hydrodynamic Model with Application to Laizhou Bay

“…It can be shown that once the water depth at any node is nonpositive at a time step, the reconstructed water depth will no longer be negative by using this method. After reconstruction, the semidry elements are further divided into two parts [41,42]: flood elements for η max ≤ z b,max and dam-break elements for η max > z b,max . Finally, the elements are classified into four types: wet elements, dry elements, flood elements, and dam-break elements.…”

“…In this model, the horizontal diffusion is not calculated at the WD interface, considering that the effects seem to be minor. For the flood type, gravity g is set to zero to ignore the influence of hydrostatic pressure in the convection term and bottom topography term to guarantee a well-balanced property [42].…”