@Article{BIngham2d,
  author  = {Fern{\'a}ndez Nieto, E. D. and Gallardo, Jos{\'e}-M. and Vigneaux, Paul},
  title   = {{E}fficient numerical schemes for viscoplastic avalanches. {P}art 2: the 2{D} case},
  journal = {Journal of Computational Physics},
  year    = {2017},
  volume  = {Accepted},
}

@Article{RMHD,
  author  = {Castro D{\'i}az, Manuel J. and Gallardo, Jos{\'e}-M. and Marquina, Antonio},
  title   = {{J}acobian-free approximate solvers for hyperbolic systems: {A}pplication to relativistic magnetohydrodynamics},
  journal = {Computer Physics Communications},
  year    = {2017},
  volume  = {219},
  pages   = {108-120},
}

@Article{Approx.Osher,
  author   = {Castro D{\'i}az, Manuel J. and Gallardo, Jos{\'e}-M. and Marquina, Antonio},
  title    = {{A}pproximate {O}sher-{S}olomon schemes for hyperbolic systems},
  journal  = {Applied Mathematics and Computation},
  year     = {2016},
  volume   = {272},
  number   = {2},
  pages    = {347-368},
  month    = {January},
  abstract = {This paper is concerned with a new kind of Riemann solvers for hyperbolic systems, which can be applied both in the conservative and nonconservative cases. In particular, the proposed schemes constitute a simple version of the classical Osher–Solomon Riemann solver, and extend in some sense the schemes proposed in Dumbser and Toro (2011) [19,20]. The viscosity matrix of the numerical flux is constructed as a linear combination of functional evaluations of the Jacobian of the flux at several quadrature points. Some families of functions have been proposed to this end: Chebyshev polynomials and rational-type functions. Our schemes have been tested with different initial value Riemann problems for ideal gas dynamics, magnetohydrodynamics and multilayer shallow water equations. The numerical tests indicate that the proposed schemes are robust, stable and accurate with a satisfactory time step restriction, and provide an efficient alternative for approximating time-dependent solutions in which the spectral decomposition is computationally expensive.},
  keywords = {Hyperbolic systems, incomplete Riemann solvers, Osher-Solomon method, Euler equations, ideal magnetohydrodynamics, multilayer shallow water equations},
  url      = {http://www.sciencedirect.com/science/article/pii/S0096300315008851},
}

This paper is concerned with a new kind of Riemann solvers for hyperbolic systems, which can be applied both in the conservative and nonconservative cases. In particular, the proposed schemes constitute a simple version of the classical Osher–Solomon Riemann solver, and extend in some sense the schemes proposed in Dumbser and Toro (2011) [19,20]. The viscosity matrix of the numerical flux is constructed as a linear combination of functional evaluations of the Jacobian of the flux at several quadrature points. Some families of functions have been proposed to this end: Chebyshev polynomials and rational-type functions. Our schemes have been tested with different initial value Riemann problems for ideal gas dynamics, magnetohydrodynamics and multilayer shallow water equations. The numerical tests indicate that the proposed schemes are robust, stable and accurate with a satisfactory time step restriction, and provide an efficient alternative for approximating time-dependent solutions in which the spectral decomposition is computationally expensive.

@Article{RVM,
  author  = {Castro D{\'i}az, Manuel J. and Gallardo, Jos{\'e}-M. and Marquina, Antonio},
  title   = {{A} class of incomplete {R}iemann solvers based on uniform rational approximations to the absolute value function},
  journal = {Journal of Scientific Computing},
  year    = {2014},
  volume  = {60},
  pages   = {363-389},
}

@Article{Avalanches-1d,
  author   = {Fern{\'a}ndez Nieto, E. D. and Gallardo, Jos{\'e}-M. and Vigneaux, Paul},
  title    = {{E}fficient numerical schemes for viscoplastic avalanches. {P}art 1: {T}he 1{D} case},
  journal  = {Journal of Computational Physics},
  year     = {2014},
  volume   = {264},
  number   = {1},
  pages    = {55-90},
  month    = {May},
  abstract = {This paper deals with the numerical resolution of a shallow water viscoplastic flow model. Viscoplastic materials are characterized by the existence of a yield stress: below a certain critical threshold in the imposed stress, there is no deformation and the material behaves like a rigid solid, but when that yield value is exceeded, the material flows like a fluid. In the context of avalanches, it means that after going down a slope, the material can stop and its free surface has a non-trivial shape, as opposed to the case of water (Newtonian fluid). The model involves variational inequalities associated with the yield threshold: finite-volume schemes are used together with duality methods (namely Augmented Lagrangian and Berm{\'u}dez–Moreno) to discretize the problem. To be able to accurately simulate the stopping behavior of the avalanche, new schemes need to be designed, involving the classical notion of well-balancing. In the present context, it needs to be extended to take into account the viscoplastic nature of the material as well as general bottoms with wet/dry fronts which are encountered in geophysical geometries. We derived such schemes and numerical experiments are presented to show their performances.},
  keywords = { Viscoplastic; Shallow water; Well-balanced; Finite volume; Variational inequality; Bingham},
}

This paper deals with the numerical resolution of a shallow water viscoplastic flow model. Viscoplastic materials are characterized by the existence of a yield stress: below a certain critical threshold in the imposed stress, there is no deformation and the material behaves like a rigid solid, but when that yield value is exceeded, the material flows like a fluid. In the context of avalanches, it means that after going down a slope, the material can stop and its free surface has a non-trivial shape, as opposed to the case of water (Newtonian fluid). The model involves variational inequalities associated with the yield threshold: finite-volume schemes are used together with duality methods (namely Augmented Lagrangian and Bermúdez–Moreno) to discretize the problem. To be able to accurately simulate the stopping behavior of the avalanche, new schemes need to be designed, involving the classical notion of well-balancing. In the present context, it needs to be extended to take into account the viscoplastic nature of the material as well as general bottoms with wet/dry fronts which are encountered in geophysical geometries. We derived such schemes and numerical experiments are presented to show their performances.

@Conference{Bingham1,
  author    = {Gallardo, Jos{\'e}-M. and Fern{\'a}ndez Nieto, E. D. and Vigneaux, Paul},
  title     = {{A} finite volume/duality method for {B}ingham viscoplastic flow},
  booktitle = {Numerical methods for hyperbolic problems: Theory and applications},
  year      = {2013},
  editor    = {V{\'a}zquez-Cend{\'o}n, Elena and Hidalgo, Arturo and Garc{\'i}a-Navarro, Pilar and Cea, Luis},
  pages     = {121-126},
  publisher = {CRC Press/Balkema},
}

@Article{castro2011comptes,
  author   = {Castro D{\'i}az, Manuel J. and Ortega-Acosta, Sergio and de la Asunci{\'o}n, Marc and Mantas, Jos{\'e} Miguel and Gallardo, Jos{\'e}-M.},
  title    = {{GPU} computing for shallow water flow simulation based on finite volume schemes},
  journal  = {Comptes Rendus M{\'e}canique},
  year     = {2011},
  volume   = {339},
  number   = {2},
  pages    = {165-184},
  abstract = {This article is a review of the work that we are carrying out to efficiently simulate shallow water flows. In this paper, we focus on the efficient implementation of path-conservative Roe type high-order finite volume schemes to simulate shallow flows that are supposed to be governed by the one-layer or two-layer shallow water systems, formulated under the form of a conservation law with source terms. The implementation of the scheme is carried out on Graphics Processing Units (GPUs), thus achieving a substantial improvement of the speedup with respect to normal CPUs. Finally, some numerical experiments are presented. },
}

This article is a review of the work that we are carrying out to efficiently simulate shallow water flows. In this paper, we focus on the efficient implementation of path-conservative Roe type high-order finite volume schemes to simulate shallow flows that are supposed to be governed by the one-layer or two-layer shallow water systems, formulated under the form of a conservation law with source terms. The implementation of the scheme is carried out on Graphics Processing Units (GPUs), thus achieving a substantial improvement of the speedup with respect to normal CPUs. Finally, some numerical experiments are presented.

@Article{gallardo2011compact,
  author   = {Gallardo, Jos{\'e}-M. and Ortega-Acosta, Sergio and de la Asunci{\'o}n, Marc and Mantas-Ruiz, Jos{\'e}-Miguel},
  title    = {{T}wo-{D}imensional {C}ompact {T}hird-{O}rder {P}olynomial {R}econstructions. {S}olving {N}onconservative {H}yperbolic {S}ystems {U}sing {GPU}s},
  journal  = {Journal of Scientific Computing},
  year     = {2011},
  volume   = {48},
  number   = {1},
  pages    = {141-163},
  abstract = {We present a new kind of high-order reconstruction operator of polynomial type, which is used in combination with the scheme presented in Castro et al. (J. Sci. Comput. 39:67–114, 2009 ) for solving nonconservative hyperbolic systems. The implementation of the scheme is carried out on Graphics Processing Units (GPUs), thus achieving a substantial improvement of the speedup with respect to normal CPUs. As an application, the two-dimensional shallow water equations with geometrical source term due to the bottom slope is considered.
},
  url      = {http://dx.doi.org/10.1007/s10915-011-9470-x},
}

We present a new kind of high-order reconstruction operator of polynomial type, which is used in combination with the scheme presented in Castro et al. (J. Sci. Comput. 39:67–114, 2009 ) for solving nonconservative hyperbolic systems. The implementation of the scheme is carried out on Graphics Processing Units (GPUs), thus achieving a substantial improvement of the speedup with respect to normal CPUs. As an application, the two-dimensional shallow water equations with geometrical source term due to the bottom slope is considered.

@Conference{HYP2008,
  author    = {Gallardo, Jos{\'e}-M. and Castro D{\'i}az, Manuel J. and Par{\'e}s, Carlos},
  title     = {{H}igh-order finite volume schemes for shallow water equations with topography and dry areas},
  booktitle = {Hyperbolic problems: Theory, numerics and applications},
  year      = {2008},
  editor    = {Eitan, Tadmor and Jian-Guo, Liu and Athanasios, Tzavaras},
  pages     = {585-594},
  publisher = {American Mathematical Society},
}

@Article{MR2383221,
  author   = {Castro D{\'i}az, Manuel J. and Gallardo, Jos{\'e}-M. and L{\'o}pez-Garc{\'i}a, Juan-A. and Par{\'e}s, Carlos},
  title    = {{W}ell-balanced high order extensions of {G}odunov's method for semilinear balance laws},
  journal  = {SIAM J. Numer. Anal.},
  year     = {2008},
  volume   = {46},
  number   = {2},
  pages    = {1012–1039},
  abstract = {This paper is concerned with the development of well-balanced high order numerical schemes for systems of balance laws with a linear flux function, whose coefficients may be variable. First, well-balanced first order numerical schemes are obtained based on the use of exact solvers of Riemann problems that include both the flux and the source terms. Godunov's methods so obtained are extended to higher order schemes by using a technique of reconstruction of states. The main contribution of this paper is to introduce a reconstruction technique that preserves the well-balanced property of Godunov's methods. Some numerical experiments are presented to verify in practice the properties of the developed numerical schemes.},
  url      = {http://epubs.siam.org/sinum/resource/1/sjnaam/v46/i2/p1012_s1},
}

This paper is concerned with the development of well-balanced high order numerical schemes for systems of balance laws with a linear flux function, whose coefficients may be variable. First, well-balanced first order numerical schemes are obtained based on the use of exact solvers of Riemann problems that include both the flux and the source terms. Godunov's methods so obtained are extended to higher order schemes by using a technique of reconstruction of states. The main contribution of this paper is to introduce a reconstruction technique that preserves the well-balanced property of Godunov's methods. Some numerical experiments are presented to verify in practice the properties of the developed numerical schemes.

@Article{MR2361537,
  author  = {Gallardo, Jos{\’e}-M. and Par{\'e}s, Carlos and Castro D{\'i}az, Manuel J.},
  title   = {{O}n a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas},
  journal = {J. Comput. Phys.},
  year    = {2007},
  volume  = {227},
  number  = {1},
  pages   = {574–601},
}

@Article{MR2219021,
  author  = {Castro D{\'i}az, Manuel J. and Gallardo, Jos{\'e}-M. and Par{\'e}s, Carlos},
  title   = {{H}igh order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. {A}pplications to shallow-water systems},
  journal = {Math. Comp.},
  year    = {2006},
  volume  = {75},
  number  = {255},
  pages   = {1103–1134},
}

@Conference{HYP2006,
  author    = {Gallardo, Jos{\’e}-M. and Castro D{\'i}az, Manuel J. and Par{\'e}s, Carlos and Gonz{\'a}lez-Vida, J.-M.},
  title     = {{O}n a well-balanced high-order finite volume scheme for the shallow water equations with bottom topography and dry areas},
  booktitle = {Hyperbolic problems: Theory, numerics and applications},
  year      = {2006},
  editor    = {Sylvie Benzoni-Gavage, Denis Serre},
  pages     = {259--270},
  publisher = {Springer},
}

@Article{MR2135784,
  author  = {Gallardo, Jos{\'e}-M. and Castro D{\'i}az, Manuel J. and Par{\'e}s, Carlos},
  title   = {{A} generalized duality method for solving variational inequalities. {A}pplications to some nonlinear {D}irichlet problems},
  journal = {Numer. Math.},
  year    = {2005},
  volume  = {100},
  number  = {2},
  pages   = {259–291},
}

@Article{ODE5,
  author  = {Gallardo, Jos{\'e}-M.},
  title   = {{G}eneration of analytic semigroups by differential operators with mixed boundary conditions},
  journal = {Rocky Mountain Journal of Mathematics},
  year    = {2003},
  volume  = {33},
  number  = {3},
  pages   = {831-863},
}

@Article{ODE4,
  author  = {Gallardo, Jos{\'e}-M.},
  title   = {{D}ifferential operators with mixed boundary conditions: generation of analytic semigroups},
  journal = {Nonlinear Analysis: Theory, Methods and Applications},
  year    = {2001},
  volume  = {47},
  pages   = {1333-1344},
}

@Article{ODE2,
  author  = {Gallardo, Jos{\'e}-M.},
  title   = {{G}eneration of analytic semigroups by second-order differential operators with non-separated boundary conditions},
  journal = {Rocky Mountain Journal of Mathematics},
  year    = {2000},
  volume  = {30},
  number  = {3},
  pages   = {869-899},
}

@Article{ODE3,
  author  = {Gallardo, Jos{\'e}-M.},
  title   = {{S}econd-order differential operators with integral boundary conditions and generation of analytic semigroups},
  journal = {Rocky Mountain Journal of Mathematics},
  year    = {2000},
  volume  = {30},
  number  = {4},
  pages   = {1265-1291},
}

@Article{ODE1,
  author  = {Gallardo, Jos{\'e}-M.},
  title   = {{S}econd-order {ODE}'s with non-separated boundary conditions and generation of analytic semigroups},
  journal = {Nonlinear Analysis: Theory, Methods and Applications},
  year    = {1997},
  volume  = {30},
  pages   = {4991-4994},
}