Non-Hydrostatic Discontinuous/Continuous Galerkin Model for Wave Propagation, Breaking and Runup

This paper presents a new depth-integrated non-hydrostatic finite element model for simulating wave propagation, breaking and runup using a combination of discontinuous and continuous Galerkin methods. The formulation decomposes the depth-integrated non-hydrostatic equations into hydrostatic and non-hydrostatic parts. The hydrostatic part is solved with a discontinuous Galerkin finite element method to allow the simulation of discontinuous flows, wave breaking and runup. The non-hydrostatic part led to a Poisson type equation, where the non-hydrostatic pressure is solved using a continuous Galerkin method to allow the modeling of wave propagation and transformation. The model uses linear quadrilateral finite elements for horizontal velocities, water surface elevations and non-hydrostatic pressures approximations. A new slope limiter for quadrilateral elements is developed. The model is verified and validated by a series of analytical solutions and laboratory experiments.

A New Method for Wet-Dry Front Treatment in Outburst Flood Simulation

When utilizing a finite volume method to predict outburst flood evolution in real geometry, the processing of wet-dry front and dry cells is an important step. In this paper, we propose a new approach to process wet-dry front and dry cells, including four steps: (1) estimating intercell properties; (2) modifying interface elevation; (3) calculating dry cell elevations by averaging intercell elevations; and (4) changing the value of the first term of slope limiter based on geometry in dry cells. The Harten, Lax, and van Leer with the contact wave restored (HLLC) scheme was implemented to calculate the flux. By combining the MUSCL (Monotone Upstream–centred Scheme for Conservation Laws)-Hancock method with the minmod slope limiter, we achieved second-order accuracy in space and time. This approach is able to keep the conservation property (C-property) and the mass conservation of complex bed geometry. The results of numerical tests in this study are consistent with experimental data, which verifies the effectiveness of the new approach. This method could be applied to acquire wetting and drying processes during flood evolution on structured meshes. Furthermore, a new settlement introduces few modification steps, so it could be easily applied to matrix calculations. The new method proposed in this study can facilitate the simulation of flood routing in real terrain.

Entropy stabilization and property-preserving limiters for ℙ1 discontinuous Galerkin discretizations of scalar hyperbolic problems

The methodology proposed in this paper bridges the gap between entropy stable and positivity-preserving discontinuous Galerkin (DG) methods for nonlinear hyperbolic problems. The entropy stability property and, optionally, preservation of local bounds for cell averages are enforced using flux limiters based on entropy conditions and discrete maximum principles, respectively. Entropy production by the (limited) gradients of the piecewise-linear DG approximation is constrained using Rusanov-type entropy viscosity. The Taylor basis representation of the entropy stabilization term reveals that it penalizes the solution gradients in a manner similar to slope limiting and requires implicit treatment to avoid severe time step restrictions. The optional application of a vertex-based slope limiter constrains the DG solution to be bounded by local maxima and minima of the cell averages. Numerical studies are performed for two scalar two-dimensional test problems with nonlinear and nonconvex flux functions.