A limiter-based well-balanced discontinuous Galerkin method for shallow-water flows with wetting and drying: Triangular grids

A novel wetting and drying treatment for second-order Runge-Kutta discontinuous Galerkin (RKDG2) methods solving the non-linear shallow water equations is proposed. It is developed for general conforming two-dimensional triangular meshes and utilizes a slope limiting strategy to accurately model inundation. The method features a non-destructive limiter, which concurrently meets the requirements for linear stability and wetting and drying. It further combines existing approaches for positivity preservation and well-balancing with an innovative velocity-based limiting of the momentum. This limiting controls spurious velocities in the vicinity of the wet/dry interface. It leads to a computationally stable and robust scheme -- even on unstructured grids -- and allows for large time steps in combination with explicit time integrators. The scheme comprises only one free parameter, to which it is not sensitive in terms of stability. A number of numerical test cases, ranging from analytical tests to near-realistic laboratory benchmarks, demonstrate the performance of the method for inundation applications. In particular, super-linear convergence, mass-conservation, well-balancedness, and stability are verified

Flood propagation modelling with the Local Inertia Approximation: theoretical and numerical analysis of its physical limitations

Attention of the researchers has increased towards a simplification of the complete Shallow water Equations called the Local Inertia Approximation (LInA), which is obtained by neglecting the advection term in the momentum conservation equation. In the present paper it is demonstrated that a shock is always developed at moving wetting-drying frontiers, and this justifies the study of the Riemann problem on even and uneven beds. In particular, the general exact solution for the Riemann problem on horizontal frictionless bed is given, together with the exact solution of the non-breaking wave propagating on horizontal bed with friction, while some example solution is given for the Riemann problem on discontinuous bed. From this analysis, it follows that drying of the wet bed is forbidden in the LInA model, and that there are initial conditions for which the Riemann problem has no solution on smoothly varying bed. In addition, propagation of the flood on discontinuous sloping bed is impossible if the bed drops height have the same order of magnitude of the moving-frontier shock height. Finally, it is found that the conservation of the mechanical energy is violated. It is evident that all these findings pose a severe limit to the application of the model. The numerical analysis has proven that LInA numerical models may produce numerical solutions, which are unreliable because of mere algorithmic nature, also in the case that the LInA mathematical solutions do not exist. The applicability limits of the LInA model are discouragingly severe, even if the bed elevation varies continuously. More important, the non-existence of the LInA solution in the case of discontinuous topography and the non-existence of receding fronts radically question the viability of the LInA model in realistic cases. It is evident that classic SWE models should be preferred in the majority of the practical applications

The VOLNA code for the numerical modelling of tsunami waves: generation, propagation and inundation

A novel tool for tsunami wave modelling is presented. This tool has the potential of being used for operational purposes: indeed, the numerical code \VOLNA is able to handle the complete life-cycle of a tsunami (generation, propagation and run-up along the coast). The algorithm works on unstructured triangular meshes and thus can be run in arbitrary complex domains. This paper contains the detailed description of the finite volume scheme implemented in the code. The numerical treatment of the wet/dry transition is explained. This point is crucial for accurate run-up/run-down computations. Most existing tsunami codes use semi-empirical techniques at this stage, which are not always sufficient for tsunami hazard mitigation. Indeed the decision to evacuate inhabitants is based on inundation maps which are produced with this type of numerical tools. We present several realistic test cases that partially validate our algorithm. Comparisons with analytical solutions and experimental data are performed. Finally the main conclusions are outlined and the perspectives for future research presented.Comment: 47 pages, 27 figures. Other author's papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh

Spectral/hp element methods: recent developments, applications, and perspectives

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate C0-continuous expansions. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed

CFD prediction and validation of ship-bank interaction in a canal

This paper utilizes CFD (Computational Fluid Dynamics) methods to investigate the bank effects on a tanker moving straight ahead at low speed in a canal characterized by surface piercing banks. For varying water depths and ship-to-bank distances, the sinkage and trim as well as the viscous hydrodynamic forces on the hull are predicted mainly by a steady state RANS (Reynolds Averaged Navier-Stokes) solver, in which the double model approximation is adopted to simulate the flat free surface. A potential flow method is also applied to evaluate the effect of the free surface and viscosity on the solutions. In addition, focus is placed on V&V (Verification and Validation) based on a grid convergence study and comparison with EFD (Experimental Fluid Dynamics) data, as well as the exploration of the modelling error in RANS computations to enable more accurate and reliable predictions of the bank effects

Moment Approximations and Model Cascades for Shallow Flow

Shallow flow models are used for a large number of applications including weather forecasting, open channel hydraulics and simulation-based natural hazard assessment. In these applications the shallowness of the process motivates depth-averaging. While the shallow flow formulation is advantageous in terms of computational efficiency, it also comes at the price of losing vertical information such as the flow's velocity profile. This gives rise to a model error, which limits the shallow flow model's predictive power and is often not explicitly quantifiable. We propose the use of vertical moments to overcome this problem. The shallow moment approximation preserves information on the vertical flow structure while still making use of the simplifying framework of depth-averaging. In this article, we derive a generic shallow flow moment system of arbitrary order starting from a set of balance laws, which has been reduced by scaling arguments. The derivation is based on a fully vertically resolved reference model with the vertical coordinate mapped onto the unit interval. We specify the shallow flow moment hierarchy for kinematic and Newtonian flow conditions and present 1D numerical results for shallow moment systems up to third order. Finally, we assess their performance with respect to both the standard shallow flow equations as well as with respect to the vertically resolved reference model. Our results show that depending on the parameter regime, e.g. friction and slip, shallow moment approximations significantly reduce the model error in shallow flow regimes and have a lot of potential to increase the predictive power of shallow flow models, while keeping them computationally cost efficient