M. J. Castro Díaz

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...

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.

A HLLC scheme for Ripa model

We consider the one-dimensional system of shallow-water equations with horizontal temperature gradients (the Ripa system). We derive a HLLC scheme for Ripa system which falls into the theory of path-conservative approximate Riemann solvers. The resulting scheme is robust, easy to implement, well-balanced, positivity preserving and entropy dissipative for the case of flat or continuous bottom.

Reliability of first order numerical schemes for solving shallow water system over abrupt topography

We compare some first order well-balanced numerical schemes for shallow water system with special interest in applications where there are abrupt variations of the topography. We show that the space step required to obtain a prescribed error depends on the method. Moreover, the solutions given by the numerical scheme can be significantly different if not enough space resolution is used. We shall pay special attention to the well-known hydrostatic reconstruction technique where it is shown that the effect of large bottom discontinuities might be missed and a modification is proposed to avoid this problem.

A second order PVM flux limiter method. Application to magnetohydrodynamics and shallow stratified flows

In this work we propose a second order flux limiter finite volume method, named PVM-2U-FL, that only uses information of the two external waves of the hyperbolic system. This method could be seen as a natural extension of the well known WAF method introduced by E.F. Toro in [23]. We prove that independently of the number of unknowns of the 1D system, it recovers the second order accuracy at regular zones, while in presence of discontinuities, the scheme degenerates to PVM-2U method, which can be seen as an improvement of the HLL method (see [6,10]). Another interesting property of the method is that it does not need any spectral decomposition of the Jacobian or Roe matrix associated to the flux function. Therefore, it can be easily applied to systems with a large number of unknowns or in situations where no analytical expression of the eigenvalues or eigenvectors are known. In this work, we apply the proposed method to magnetohydrodynamics and to stratified multilayer flows. Comparison with the two-waves WAF and HLL-MUSCL methods are also presented. The numerical results show that PVM-2U-FL is the most efficient and accurate among them.

A Duality Method for Sediment Transport Based on a Modified Meyer-Peter & Müller Model

This article focuses on the simulation of the sediment transport by a fluid in contact with a sediment layer. This phenomena can be modelled by using a coupled model constituted by a hydrodynamical component, described by a shallow water system, and a morphodynamical one, which depends on a solid transport flux given by some empirical law. The solid transport discharge proposed by Meyer-Peter & Müller is one of the most popular but it has the inconvenient of not including pressure forces. Due to this, this formula produces numerical simulations that are not realistic in zones where gravity effects are relevant, e.g. advancing front of the sand layer. Moreover, the thickness of the sediment layer is not taken into account and, as a consequence, mass conservation of sediment may fail. Fowler et al. proposed a generalization that takes into account gravity effects as well as the thickness of the sediment layer which is in better agreement with the physics of the problem. We propose to solve this system by using a path-conservative scheme for the hydrodynamical part and a duality method based on Bermúdez-Moreno algorithm for the morphodynamical component.

Convergence of Path-Conservative Numerical Schemes for Hyperbolic Systems of Balance Laws

We are concerned with the numerical approximation of Cauchy problems for hyperbolic systems of balance laws, which can be studied as a particular case of nonconservative hyperbolic systems. We consider the theory developed by Dal Maso, LeFloch, and Murat to define the weak solutions of nonconservative systems, and path-conservative numerical schemes (introduced by Pares) to numerically approximate these solutions. In a previous work with Le Floch we have studied the appearance of a convergence error measure in the general case of noconservative hyperbolic systems, and we have noticed that this lack of convergence cannot always be observed in numerical experiments. In this work we study the convergence of path-conservative schemes for the special case of systems of balance laws, specifically, the experiments performed up to now show that the numerical solutions converge to the right weak solutions for the correct choice of path-conservative scheme.

On a Sediment Transport Model in Shallow Water Equations with Gravity Effects

Sediment transport by a fluid over a sediment layer can be modeled by a coupled system with a hydrodynamical component, described by a shallow water system, and a morphodynamical component, given by a solid transport flux. Meyer-Peter and Muller developed one of the most known formulae for solid transport discharge, but it has the inconvenient of not including pressure forces. This makes numerical simulations not accurate in zones where gravity effects are relevant, e.g., advancing front of the sand layer. Fowler et al. proposed a generalization that takes into account gravity effects as well as the length of the sediment layer which agrees better to the physics of the problem. We propose to solve this system by using a path-conservative scheme for the hydrodynamical part and a duality method based on Bermudez-Moreno algorithm for the morphodynamical component.

A High Order Finite Volume Numerical Scheme for Shallow Water System: An Efficient Implementation on GPUs

In this work we present a high order finite volume numerical scheme for solving the one layer shallow-water system. The numerical solution of this model is useful for several applications related to geophysical flows, and they impose a great demand of computing power. As a consequence, extremely efficient high performance solvers are required. In this work we perform a GPU implementation of the proposed numerical scheme and some computations are made to test the performance of the implementation.

A High-Order Finite Volume Method for Nonconservative Problems and Its Application to Model Submarine Avalanches

In this chapter we investigate how to apply a high-order finite volume method to discretize the model proposed in [FeBo08] to study submarine avalanches.