Sur la solution numerique des systems creuses et linéaires émergents de la discretization volume finis des modeles 2D de type Boussinesq

This work supplements the realization and validation of a higher-order well balanced finite volume (FV) scheme developed for numerically simulating, on triangular meshes, weakly non-linear weakly dispersive water waves over varying bathymetries. The scheme has been recently presented by Kazolea et al. \textit{(Coastal Eng. 69:42-66, 2012 and J. Comp. Phys. 271:281-305, 2014)}. More precisely, we investigate and develop solution strategies for the sparse linear system that occurs during this FV discretisation of a set of Boussinesq-type equations on unstructured meshes. The resultant system of equations must be solved at each time step as to recover the actual velocity field of the flow. The system's coefficient matrix is sparse, un-symmetric and often ill-conditioned. Its characteristics are affected by physical quantities of the problem to be solved, such as the un-disturbed water depth and the mesh topology. This work investigates the application of different iterative techniques, with and without the usage of preconditioners and reordering, for the solution of this sparse linear system. Two different iterative methods, three preconditioning techniques, including different ILU factorizations and two different reordering techniques are implemented and discussed. An optimal strategy, in terms of computational efficiency and robustness, is proposed.Ce travail concerne la réalisation et la validation d’un schéma Volumes Finis d’ordre élevé pour la simulation des vagues en régime faiblement non-linéaire et faiblement dispersife sur bathymétries variables. Le schéma implémenté est celui proposé récemment par Kazolea et al. (Coastal Eng 69:. 42-66, 2012 et J. Phys Comp 271:.. 281-305, 2014). Plus précisément, nous étudions et développons des stratégies de solution pour le système linéaire creux qui se produit au cours de la discrétisation des équations de Boussinesq sur maillagesnon structurés. Le système d’équations résultant doit être résolu à chaque pas de temps pour récupérer la vitesse. La matrice du système est creuse, non symétrique et souvent mal conditionné. Ses caractéristiques sont affectées par des quantités physiques tels que la profondeur de l’ eau au repos et la topologie du maillage. Ce travail étudie l’ application de différentes techniques itératives, avec et sans l’ utilisation de pré conditionneurs et de ré-numérotation, pour la solution de ce système linéaire creux. Deux méthodes itératives différentes, troistechniques de pré conditionnement, y compris les différents factorisations ILU et deux techniques de ré ordonnancement différentes sont mises en œuvre et évaluées. Une stratégie optimale, en termes d’efficacité de calcul et de robustesse, est proposé

Advanced Numerical Simulation of Near-shore processes by Extended Boussinesq-Type Models on Unstructured Meshes

International audienc

An unstructured finite volume numerical scheme for extended Boussinesq-type equations for irregular wave propagation

The interplay between low and high frequency waves is groundwork for the near-shore hydrodynamics for which Boussinesq-type (BT) equations are widely applied dur- ing the past few decades to model the waves’s propagation and transformations. In this work, the TUCWave code is vali- dated with respect to the propagation, transformation, breaking and run-up of irregular waves. The main aim is to investigate the ability of the model and the breaking wave parametriza- tions used in the code to reproduce the nonlinear properties of the waves in the surf zone. The TUCWave code numer- ically solves the 2D BT equations of Nwogu (1993) on un- structured meshes, using a novel high-order well-balanced fi- nite volume (FV) numerical scheme following the median dual vertex-centered approach. The BT equations are recast in the form of a system of conservation laws and the conservative FV scheme developed is of the Godunov-type. The approxi- mate Riemann solver of Roe for the advective fluxes is utilized along with a well-balanced topography source term upwinding and accurate numerical treatment of moving wet/dry fronts. The dispersion terms are discretized using a consistent, to the FV framework, discretization and the friction stresses are also included. High-order spatial accuracy is achieved through a MUSCL-type reconstruction technique and temporal through a strong stability preserving Runge-Kutta time stepping. Wave breaking mechanism have also been developed and incorpo- rated into the model. TUCWave code is applied to bench- mark test cases and real case scenarios where the shoaling and breaking of irregular waves is investigated

On the numerical solution of sparse linear systems emerging in finite volume discretizations of 2D Boussinesq-type models

Ce travail concerne la réalisation et la validation d’un schéma Volumes Finis d’ordre élevé pour la simulation des vagues en régime faiblement non-linéaire et faiblement dispersife sur bathymétries variables. Le schéma implémenté est celui proposé récemment par Kazolea et al. (Coastal Eng 69:. 42-66, 2012 et J. Phys Comp 271:.. 281-305, 2014). Plus précisément, nous étudions et développons des stratégies de solution pour le système linéaire creux qui se produit au cours de la discrétisation des équations de Boussinesq sur maillagesnon structurés. Le système d’équations résultant doit être résolu à chaque pas de temps pour récupérer la vitesse. La matrice du système est creuse, non symétrique et souvent mal conditionné. Ses caractéristiques sont affectées par des quantités physiques tels que la profondeur de l’ eau au repos et la topologie du maillage. Ce travail étudie l’ application de différentes techniques itératives, avec et sans l’ utilisation de pré conditionneurs et de ré-numérotation, pour la solution de ce système linéaire creux. Deux méthodes itératives différentes, troistechniques de pré conditionnement, y compris les différents factorisations ILU et deux techniques de ré ordonnancement différentes sont mises en œuvre et évaluées. Une stratégie optimale, en termes d’efficacité de calcul et de robustesse, est proposé.This work supplements the realization and validation of a higher-order well balanced finite volume (FV) scheme developed for numerically simulating, on triangular meshes, weakly non-linear weakly dispersive water waves over varying bathymetries. The scheme has been recently presented by Kazolea et al. \textit{(Coastal Eng. 69:42-66, 2012 and J. Comp. Phys. 271:281-305, 2014)}. More precisely, we investigate and develop solution strategies for the sparse linear system that occurs during this FV discretisation of a set of Boussinesq-type equations on unstructured meshes. The resultant system of equations must be solved at each time step as to recover the actual velocity field of the flow. The system's coefficient matrix is sparse, un-symmetric and often ill-conditioned. Its characteristics are affected by physical quantities of the problem to be solved, such as the un-disturbed water depth and the mesh topology. This work investigates the application of different iterative techniques, with and without the usage of preconditioners and reordering, for the solution of this sparse linear system. Two different iterative methods, three preconditioning techniques, including different ILU factorizations and two different reordering techniques are implemented and discussed. An optimal strategy, in terms of computational efficiency and robustness, is proposed

Robust code-to-code comparison for long wave run up

International audienc

Runup and uncertainty quantification: sensitivity analysis via ANOVA decomposition

We investigate the ability of uncertainty quantification techniques to act as enablers for the study of the sensitivity of dynamics of dam breaks to the variations of model parameters. In particular, we make use of sensitivity indexes computed by means of an Analysis of Variance (ANOVA) to provide the sensitivity of the runup dynamics to the variations of parameters such wave amplitude, friction coefficient, etc. The sensitivity indexes, known as Sobol indexes, are obtained following (Crestaux-LeMaitre-Martinez, 2009) by resorting to a non-intrusive polynomial chaos method allowing to reconstruct a complete representation of the variation of the outputs in the parameter space, and to compute the sensitivity indexes via the ANOVA decomposition. To increase the reliability of the results, we perfom the study independently with two models based on a discretization of the shallow water equations, developed in (Ricchiuto, 2014), and (Nikolos and Delis, 2009), respectively. The approach proposed provides simultaneously the variance of the outputs and their sensitivity to each independent parameter, allowing to construct a hierarchy of parameters which depends on the flow conditions

Numerical study of wave conditions for the old Venetian harbor of Chania in Crete, Greece

In this work we present a numerical study of the long wave conditions induced in the old Venetian Harbor of Chania (in the island of Crete, Greece) using two models. The fully nonlinear-weakly dispersive COULWAVE code and the weakly nonlinear-weakly dispersive TUCWave code. The two models are used to determine the resonant frequencies, amplitudes and modes of the entire harbor basin. The presented results are compared and discusse