Enrique D. Fernández-Nieto

High Order Extensions of Roe Schemes for Two-Dimensional Nonconservative Hyperbolic Systems

This paper is concerned with the development of well-balanced high order Roe methods for two-dimensional nonconservative hyperbolic systems. In particular, we are interested in extending the methods introduced in (Castro et al., Math. Comput. 75:1103–1134, 2006) to the two-dimensional case. We also investigate the well-balance properties and the consistency of the resulting schemes. We focus in applications to one and two layer shallow water systems.

A Class of Computationally Fast First Order Finite Volume Solvers: PVM Methods

In this work, we present a class of fast first order finite volume solvers, called PVM (polynomial viscosity matrix), for balance laws or, more generally, for nonconservative hyperbolic systems. They are defined in terms of viscosity matrices computed by a suitable polynomial evaluation of a Roe matrix. These methods have the advantage that they only need some information about the eigenvalues of the system to be defined, and no spectral decomposition of a Roe matrix is needed. As a consequence, they are faster than the Roe method. These methods can be seen as a generalization of the schemes introduced by Degond et al. in [C. R. Acad. Sci. Paris Ser. l Math., 328 (1999), pp. 479--483] for balance laws and nonconservative systems. The first order path conservative methods to be designed here are intended to be used as the basis for higher order methods for multidimensional problems. In this work, some well known solvers, such as Rusanov, Lax--Friedrichs, FORCE (see [E. F. Toro and S. J. Billett, IMA J. Num...

Numerical Treatment of the Loss of Hyperbolicity of the Two-Layer Shallow-Water System

This article is devoted to the numerical solution of the inviscid two-layer shallow water system. This system may lose the hyperbolic character when the shear between the layer is big enough. This loss of hyperbolicity is related to the appearance of shear instabilities that leads, in real flows, to intense mixing of the two layers that the model is not able to simulate. The strategy here is to add some extra friction terms, which are supposed to parameterize the loss of mechanical energy due to mixing, to get rid of this difficulty. The main goal is to introduce a technique allowing one to add locally and automatically an ‘optimal’ amount of shear stress to make the flow to remain in the hyperbolicity region. To do this, first an easy criterium to check the hyperbolicity of the system for a given state is proposed and checked. Next, we introduce a predictor/corrector strategy. In the predictor stage, a numerical scheme is applied to the system without extra friction. In the second stage, a discrete semi-implicit linear friction law is applied at any cell in which the predicted states are not in the hyperbolicity region. The coefficient of this law is calculated so that the predicted states are driven to the boundary of the hyperbolicity region according to the proposed criterium. The numerical scheme to be used at the first stage has to be able to advance in time in presence of complex eigenvalues: we propose here a family of path-conservative numerical scheme having this property. Finally, some numerical tests have been performed to assess the efficiency of the proposed strategy.

Augmented Lagrangian Method and Compressible Visco-plastic Flows: Applications to Shallow Dense Avalanches

In this paper we propose a well-balanced finite volume/augmented Lagrangian method for compressible visco-plastic models focusing on a compressible Bingham type system with applications to dense avalanches. For the sake of completeness we also present a method showing that such a system may be derived for a shallow flow of a rigid-viscoplastic incompressible fluid, namely for incompressible Bingham type fluid with free surface. When the fluid is relatively shallow and spreads slowly, lubrication-style asymptotic approximations can be used to build reduced models for the spreading dynamics, see for instance [N.J. Balmforth et al., J. Fluid Mech (2002)]. When the motion is a little bit quicker, shallow water theory for non-Newtonian flows may be applied, for instance assuming a Navier type boundary condition at the bottom. We start from the variational inequality for an incompressible Bingham fluid and derive a shallow water type system. In the case where Bingham number and viscosity are set to zero we obtain the classical Shallow Water or Saint-Venant equations obtained for instance in [J.F. Gerbeau, B. Perthame, DCDS (2001)]. For numerical purposes, we focus on the one-dimensional in space model: We study associated static solutions with sufficient conditions that relate the slope of the bottom with the Bingham number and domain dimensions. We also propose a well-balanced finite volume/augmented Lagrangian method. It combines well-balanced finite volume schemes for spatial discretization with the augmented Lagrangian method to treat the associated optimization problem. Finally, we present various numerical tests.

An MPI-CUDA implementation of an improved Roe method for two-layer shallow water systems

The numerical solution of two-layer shallow water systems is required to simulate accurately stratified fluids, which are ubiquitous in nature: they appear in atmospheric flows, ocean currents, oil spills, etc. Moreover, the implementation of the numerical schemes to solve these models in realistic scenarios imposes huge demands of computing power. In this paper, we tackle the acceleration of these simulations in triangular meshes by exploiting the combined power of several CUDA-enabled GPUs in a GPU cluster. For that purpose, an improvement of a path conservative Roe-type finite volume scheme which is specially suitable for GPU implementation is presented, and a distributed implementation of this scheme which uses CUDA and MPI to exploit the potential of a GPU cluster is developed. This implementation overlaps MPI communication with CPU-GPU memory transfers and GPU computation to increase efficiency. Several numerical experiments, performed on a cluster of modern CUDA-enabled GPUs, show the efficiency of the distributed solver.

On an Intermediate Field Capturing Riemann Solver Based on a Parabolic Viscosity Matrix for the Two-Layer Shallow Water System

The goal of this article is to design a new approximate Riemann solver for the two-layer shallow water system which is fast compared to Roe schemes and accurate compared to Lax-Friedrichs, FORCE, or GFORCE schemes (see Castro et al. in Math. Comput. 79:1427–1472, 2010). This Riemann solver is based on a suitable decomposition of a Roe matrix (see Toumi in J. Comput. Phys. 102(2):360–373, 1992) by means of a parabolic viscosity matrix (see Degond et al. in C. R. Acad. Sci. Paris 1 328:479–483, 1999) that captures some information concerning the intermediate characteristic fields. The corresponding first order numerical scheme, which is called IFCP (Intermediate Field Capturing Parabola) is linearly L∞-stable, well-balanced, and it doesn’t require an entropy-fix technique. Some numerical experiments are presented to compare the behavior of this new scheme with Roe and GFORCE methods.

Extension of WAF Type Methods to Non-Homogeneous Shallow Water Equations with Pollutant

This paper deals with the extension of the WAF method to discretize Shallow Water Equations with pollutants. We consider two different versions of the WAF method, by approximating the intermediate waves using the flux of HLL or the direct approach of HLLC solver. It is seen that both versions can be written under the same form with different definitions for the approximation of the velocity waves. We also propose an extension of the method to non-homogeneous systems. In the case of homogeneous systems it is seen that we can rewrite the third component of the numerical flux in terms of an intermediate wave speed approximation. We conclude that—in order to have the same relation for non-homogeneous systems—the approximation of the intermediate wave speed must be modified. The proposed extension of the WAF method preserves all stationary solutions, up to second order accuracy, and water at rest in an exact way, even with arbitrary pollutant concentration. Finally, we perform several numerical tests, by comparing it with HLLC solver, reference solutions and analytical solutions.

AN ENERGETICALLY CONSISTENT VISCOUS SEDIMENTATION MODEL

In this paper we consider a two-dimensional viscous sedimentation model which is a viscous Shallow–Water system coupled with a diffusive equation that describes the evolution of the bottom. For this model, we prove the stability of weak solutions for periodic domains and give some numerical experiments. We also discuss around various discharge quantity choices.

A Multilayer Method for the Hydrostatic Navier-Stokes Equations: A Particular Weak Solution

In this work we present a multilayer approach to the solution of non-stationary 3D Navier–Stokes equations. We use piecewise smooth weak solutions. We approximate the velocity by a piecewise constant (in z) horizontal velocity and a linear (in z) vertical velocity in each layer, possibly discontinuous across layer interfaces. The multilayer approach is deduced by using the variational formulation and by considering a reduced family of test functions. The procedure naturally provides the mass and momentum interfaces conditions. The mass and momentum conservation across interfaces is formulated via normal flux jump conditions. The jump conditions associated to momentum conservation are formulated by means of an approximation of the vertical derivative of the velocity that appears in the stress tensor. We approximate the multilayer model for hydrostatic pressure, by using a polynomial viscosity matrix finite volume scheme and we present some numerical tests that show the main advantages of the model: it improves the approximation of the vertical velocity, provides good predictions for viscous effects and simulates re-circulations behind solid obstacles.

A Well-balanced Finite Volume-Augmented Lagrangian Method for an Integrated Herschel-Bulkley Model

We are interested in the derivation of an integrated Herschel-Bulkley model for shallow flows, as well as in the design of a numerical algorithm to solve the resulting equations. The goal is to simulate the evolution of thin sheet of viscoplastic materials on inclined planes and, in particular, to be able to compute the evolution from dynamic to stationary states. The model involves a variational inequality and it is valid from null to moderate slopes. The proposed numerical scheme is well balanced and involves a coupling between a duality technique (to treat plasticity), a fixed point method (to handle the power law) and a finite volume discretization. Several numerical tests are done, including a comparison with an analytical solution, to confirm the well balanced property and the ability to cope with the various rheological regimes associated with the Herschel-Bulkley constitutive law.