| Morales de Luna, T. (2008), "A Saint Venant model for gravity driven shallow water flows with variable density and compressibility effects", Mathematical and Computer Modelling., February, 2008. Vol. 47(3-4), pp. 436-444. |
| Abstract: We introduce a new model for shallow water flows with non-flat bottom made of two layers of compressible–incompressible fluids. The classical Savage–Hutter model for gravity driven shallow water flows is derived from incompressible Euler equations. Here, we generalize this model by adding an upper compressible layer. We obtain a model of shallow water type, that admits an entropy dissipation inequality, preserves the steady state of a lake at rest and gives an approximation of the free surface compressible–incompressible Euler equations. Keywords: Shallow water; Variable density; Compressibility; Entropy inequality; Savage–Hutter model |
BibTeX:
@article{moralesdeluna2008saint,
author = {Morales de Luna, Tomás},
title = {A Saint Venant model for gravity driven shallow water flows with variable density and compressibility effects},
journal = {Mathematical and Computer Modelling},
year = {2008},
volume = {47},
number = {3-4},
pages = {436--444},
url = {http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V0V-4NTJH02-6&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=c714fddce3783429707813d440feb62a},
doi = {10.1016/j.mcm.2007.04.016}
}
|
| Bouchut, F. and Morales de Luna, T. (2008), "An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment", ESAIM: Mathematical Modelling and Numerical Analysis (ESAIM: M2AN)., jun, 2008. Vol. 42, pp. 683-698. |
| Abstract: We consider the system of partial differential equations governing the one-dimensional flow of two superposed immiscible layers of shallow water. The difficulty in this system comes from the coupling terms involving some derivatives of the unknowns that make the system nonconservative, and eventually nonhyperbolic. Due to these terms, a numerical scheme obtained by performing an arbitrary scheme to each layer, and using time-splitting or other similar techniques leads to instabilities in general. Here we use entropy inequalities in order to control the stability. We introduce a stable well-balanced time-splitting scheme for the two-layer shallow water system that satisfies a fully discrete entropy inequality. In contrast with Roe type solvers, it does not need the computation of eigenvalues, which is not simple for the two-layer shallow water system. The solver has the property to keep the water heights nonnegative, and to be able to treat vanishing values. |
BibTeX:
@article{bouchut2008entropy,
author = {Bouchut, François and Morales de Luna, Tomás},
title = {An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis (ESAIM: M2AN)},
year = {2008},
volume = {42},
pages = {683--698},
url = {http://www.esaim-m2an.org/index.php?option=article&access=standard&Itemid=129&url=/articles/m2an/abs/first/m2an0716/m2an0716.html},
doi = {10.1051/m2an:2008019}
}
|
| Morales de Luna, T., Castro Díaz, M.J., Parés Madroñal, C. and Fernández Nieto, E.D. (2009), "On a shallow water model for the simulation of turbidity currents", Communications in Computational Physics. Vol. 6(4), pp. 848-882. |
| Abstract: We present a model for hyperpycnal plumes or turbidity currents that takes into account the interaction between the turbidity current and the bottom, considering deposition and erosion effects as well as solid transport of particles at the bed load due to the current. Water entrainment from the ambient water in which the turbidity current plunges is also considered. Motion of ambient water is neglected and the rigid lid assumption is considered. The model is obtained as a depth-average system of equations under the shallow water hypothesis describing the balance of fluid mass, sediment mass and mean flow. The character of the system is analyzed and numerical simulations are carried out using finite volume schemes and path-conservative Roe schemes. |
BibTeX:
@article{moralesdeluna2009shallow,
author = {Morales de Luna, Tomás and Castro Díaz, Manuel J. and Parés Madroñal, Carlos and Fernández Nieto, Enrique D.},
title = {On a shallow water model for the simulation of turbidity currents},
journal = {Communications in Computational Physics},
year = {2009},
volume = {6},
number = {4},
pages = {848--882},
url = {http://www.global-sci.com/cgi-bin/fulltext/6/848/full},
doi = {10.4208/cicp.2009.v6.p848}
}
|
| Bouchut, F. and Morales de Luna, T. (2009), "Semi-discrete Entropy Satisfying Approximate Riemann Solvers. The Case of the Suliciu Relaxation Approximation", Journal of Scientific Computing. Vol. 41, pp. 483-509. |
| Abstract: Abstract In this work we establish conditions for an approximate simple Riemann solver to satisfy a semi-discrete entropy inequality. The semi-discrete approach is less restrictive than the fully-discrete case and allows to grant some other good properties for numerical schemes. First, conditions are established in an abstract framework for simple Riemann solvers to satisfy a semi-discrete entropy inequality and then the results are applied, as a particular case, to the Suliciu system. This will lead in particular to the definition of schemes for the isentropic gas dynamics and the full gas dynamics system that are stable and preserve the stationary shocks. |
BibTeX:
@article{bouchut2009semi,
author = {Bouchut, François and Morales de Luna, Tomás},
title = {Semi-discrete Entropy Satisfying Approximate Riemann Solvers. The Case of the Suliciu Relaxation Approximation},
journal = {Journal of Scientific Computing},
year = {2009},
volume = {41},
pages = {483--509},
url = {http://dx.doi.org/10.1007/s10915-009-9311-3},
doi = {10.1007/s10915-009-9311-3}
}
|
| Bouchut, F. and Morales de Luna, T. (2010), "A Subsonic-Well-Balanced Reconstruction Scheme for Shallow Water Flows", SIAM Journal on Numerical Analysis. Vol. 48(5), pp. 1733-1758. |
| Abstract: We consider the Saint-Venant system for shallow water flows with non-flat bottom. In the past years, efficient well-balanced methods have been proposed in order to well resolve solutions close to steady states at rest. Here we describe a strategy based on a local subsonic steady-state reconstruction that allows to derive a subsonic-well-balanced scheme, preserving exactly all the subsonic steady states. It generalizes the now wellknown hydrostatic solver, and as the latter it preserves nonnegativity of water height and satisfies a semi-discrete entropy inequality. An application to the Euler-Poisson system is proposed. |
BibTeX:
@article{bouchut2010subsonic,
author = {Bouchut, Francois and Morales de Luna, Tomas},
title = {A Subsonic-Well-Balanced Reconstruction Scheme for Shallow Water Flows},
journal = {SIAM Journal on Numerical Analysis},
year = {2010},
volume = {48},
number = {5},
pages = {1733--1758},
url = {http://link.aip.org/link/SJNAAM/v48/i5/p1733/s1&Agg=doi},
doi = {10.1137/090758416}
}
|
| Morales de Luna, T., Castro Díaz, M.J. and Parés Madroñal, C. (2010), "On a sediment transport model in shallow water equations with gravity effects", Numerical Mathematics and Advanced Applications 2009. , pp. 655-662. |
| Abstract: 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 Müller 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. |
BibTeX:
@article{moralesdeluna2010sediment,
author = {Morales de Luna, Tomás and Castro Díaz, Manuel J. and Parés Madroñal, Carlos},
title = {On a sediment transport model in shallow water equations with gravity effects},
journal = {Numerical Mathematics and Advanced Applications 2009},
year = {2010},
pages = {655--662},
url = {http://www.springer.com/mathematics/computational+science+%26+engineering/book/978-3-642-11794-7},
doi = {10.1007/978-3-642-11795-4}
}
|
| Morales de Luna, T., Castro Díaz, M.J. and Parés Madroñal, C. (2011), "A Duality Method for Sediment Transport Based on a Modified Meyer-Peter & Müller Model", Journal of Scientific Computing., December, 2011. Vol. 48, pp. 258-273. |
| Abstract: 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. |
BibTeX:
@article{moralesdeluna2011duality,
author = {Morales de Luna, T. and Castro Díaz, M. J. and Parés Madroñal, C.},
title = {A Duality Method for Sediment Transport Based on a Modified Meyer-Peter & Müller Model},
journal = {Journal of Scientific Computing},
year = {2011},
volume = {48},
pages = {258-273},
url = {https://link.springer.com/article/10.1007/s10915-010-9447-1},
doi = {10.1007/s10915-010-9447-1}
}
|
| Cordier, S., Le, M.H.. and Morales de Luna, T. (2011), "Bedload transport in shallow water models: Why splitting (may) fail, how hyperbolicity (can) help", Advances in Water Resources., August, 2011. Vol. 34(8), pp. 980-989. |
| Abstract: In this paper, we are concerned with sediment transport models consisting of a shallow water system coupled with the so called Exner equation to describe the evolution of the topography. We show that, for some bedload transport models like the well-known Meyer-Peter and Müller model, the system is hyperbolic and, thus, linearly stable, only under some constraint on the velocity. In practical situations, this condition is hopefully fulfilled. Numerical approximations of such system are often based on a splitting method, solving first shallow water equation on a time step and, updating afterwards the topography. It is shown that this strategy can create spurious/unphysical oscillations which are related to the study of hyperbolicity. Using an upper bound of the largest eigenvector may improve the results although the instabilities cannot be always avoided, e.g. in supercritical regions. |
BibTeX:
@article{cordier2011bedload,
author = {Cordier, S. and Le, M.H. and Morales de Luna, T.},
title = {Bedload transport in shallow water models: Why splitting (may) fail, how hyperbolicity (can) help},
journal = {Advances in Water Resources},
year = {2011},
volume = {34},
number = {8},
pages = {980--989},
url = {http://www.sciencedirect.com/science/article/pii/S0309170811000935},
doi = {10.1016/j.advwatres.2011.05.002}
}
|
| Castro Díaz, M.J., Fernández-Nieto, E.D., Morales de Luna, T., Narbona-Reina, G. and Parés Madroñal, C. (2013), "A HLLC scheme for nonconservative hyperbolic problems. Application to turbidity currents with sediment transport", ESAIM: Mathematical Modelling and Numerical Analysis., July, 2013. Vol. 47(1), pp. 1-32. |
| Abstract: The goal of this paper is to obtain a well-balanced, stable, fast, and robust HLLC-type approximate Riemann solver for a hyperbolic nonconservative PDE system arising in a turbidity current model. The main difficulties come from the nonconservative nature of the system. A general strategy to derive simple approximate Riemann solvers for nonconservative systems is introduced, which is applied to the turbidity current model to obtain two different HLLC solvers. Some results concerning the non-negativity preserving property of the corresponding numerical methods are presented. The numerical results provided by the two HLLC solvers are compared between them and also with those obtained with a Roe-type method in a number of 1d and 2d test problems. This comparison shows that, while the quality of the numerical solutions is comparable, the computational cost of the HLLC solvers is lower, as only some partial information of the eigenstructure of the matrix system is needed. |
BibTeX:
@article{castrodiaz2013hllc,
author = {Castro Díaz, Manuel Jesús and Fernández-Nieto, Enrique Domingo and Morales de Luna, Tomás and Narbona-Reina, Gladys and Parés Madroñal, Carlos},
title = {A HLLC scheme for nonconservative hyperbolic problems. Application to turbidity currents with sediment transport},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
year = {2013},
volume = {47},
number = {1},
pages = {1--32},
url = {https://www.esaim-m2an.org/articles/m2an/abs/2013/01/m2an120017/m2an120017.html},
doi = {10.1051/m2an/2012017}
}
|
| Fernández-Nieto, E., Koné, E., Morales de Luna, T. and Bürger, R. (2013), "A multilayer shallow water system for polydisperse sedimentation", Journal of Computational Physics., April, 2013. Vol. 238, pp. 281-314. |
| Abstract: This work considers the flow of a fluid containing one disperse substance consisting of small particles that belong to different species differing in size and density. The flow is modelled by combining a multilayer shallow water approach with a polydisperse sedimentation process. This technique allows one to keep information on the vertical distribution of the solid particles in the mixture, and thereby to model the segregation of the particle species from each other, and from the fluid, taking place in the vertical direction of the gravity body force only. This polydisperse sedimentation process is described by the well-known Masliyah-Lockett-Bassoon (MLB) velocity functions. The resulting multilayer sedimentation-flow model can be written as a hyperbolic system with nonconservative products. The definitions of the nonconservative products are related to the hydrostatic pressure and to the mass and momentum hydrodynamic transfer terms between the layers. For the numerical discretization a strategy of two steps is proposed, where the first one is also divided into two parts. In the first step, instead of approximating the complete model, we approximate a reduced model with a smaller number of unknowns. Then, taking advantage of the fact that the concentrations are passive scalars in the system, we approximate the concentrations of the different species by an upwind scheme related to the numerical flux of the total concentration. In the second step, the effect of the transference terms defined in terms of the MLB model is introduced. These transfer terms are approximated by using a numerical flux function used to discretize the 1D vertical polydisperse model, see Bürger et al. [ R. Bürger, A. García, K.H. Karlsen, J.D. Towers, A family of numerical schemes for kinematic flows with discontinuous flux, J. Eng. Math. 60 (2008) 387-425]. Finally, some numerical examples are presented. Numerical results suggest that the multilayer shallow water model could be adequate in situations where the settling takes place from a suspension that undergoes horizontal movement. |
BibTeX:
@article{fernandez-nieto2013multilayer,
author = {E.D. Fernández-Nieto and E.H. Koné and Morales de Luna, Tomás and R. Bürger},
title = {A multilayer shallow water system for polydisperse sedimentation},
journal = {Journal of Computational Physics},
year = {2013},
volume = {238},
pages = {281--314},
url = {http://www.sciencedirect.com/science/article/pii/S0021999112007395},
doi = {10.1016/j.jcp.2012.12.008}
}
|
| Morales de Luna, T., Castro Díaz, M.J.. and Parés, C. (2013), "Reliability of first order numerical schemes for solving shallow water system over abrupt topography", Applied Mathematics and Computation., May, 2013. Vol. 219(17), pp. 9012-9032. |
| Abstract: Abstract 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. |
BibTeX:
@article{moralesdeluna2013reliability,
author = {Morales de Luna, T. and Castro Díaz, M.J. and Parés, C.},
title = {Reliability of first order numerical schemes for solving shallow water system over abrupt topography},
journal = {Applied Mathematics and Computation},
year = {2013},
volume = {219},
number = {17},
pages = {9012--9032},
url = {http://www.sciencedirect.com/science/article/pii/S0096300313002865},
doi = {10.1016/j.amc.2013.03.033}
}
|
| Fernández-Nieto, E.D., Lucas, C., Morales de Luna, T. and Cordier, S. (2014), "On the influence of the thickness of the sediment moving layer in the definition of the bedload transport formula in Exner systems", Computers & Fluids., March, 2014. Vol. 91, pp. 87-106. |
| Abstract: In this paper we study Exner system and introduce a modified general definition for bedload transport flux. The new formulation has the advantage of taking into account the thickness of the sediment layer which avoids mass conservation problems in certain situations. Moreover, it reduces to a classical solid transport discharge formula in the case of quasi-uniform regime. We also present several numerical tests where we compare the proposed sediment transport formula with the classical formulation and we show the behavior of the new model in different configurations. |
BibTeX:
@article{fernandez-nieto2014influence,
author = {Fernández-Nieto, E. D. and Lucas, C. and Morales de Luna, T. and Cordier, S.},
title = {On the influence of the thickness of the sediment moving layer in the definition of the bedload transport formula in Exner systems},
journal = {Computers & Fluids},
year = {2014},
volume = {91},
pages = {87--106},
url = {http://www.sciencedirect.com/science/article/pii/S0045793013004866},
doi = {10.1016/j.compfluid.2013.11.031}
}
|
| Morales de Luna, T., Castro Díaz, M.J. and Parés, C. (2014), "Relation between PVM schemes and simple Riemann solvers", Numerical Methods for Partial Differential Equations., March, 2014. Vol. 30(4), pp. 1315-1341. |
| Abstract: Approximate Riemann solvers (ARS) and polynomial viscosity matrix (PVM) methods constitute two general frameworks to derive numerical schemes for hyperbolic systems of Partial Differential Equations (PDE's). In this work, the relation between these two frameworks is analyzed: we show that every PVM method can be interpreted in terms of an approximate Riemann solver provided that it is based on a polynomial that interpolates the absolute value function at some points. Furthermore, the converse is true provided that the ARS satisfies a technical property to be specified. Besides its theoretical interest, this relation provides a useful tool to investigate the properties of some well-known numerical methods that are particular cases of PVM methods, as the analysis of some properties is easier for ARS methods. We illustrate this usefulness by analyzing the positivity-preservation property of some well-known numerical methods for the shallow water system. 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2014 |
BibTeX:
@article{moralesdeluna2014relation,
author = {Morales de Luna, Tomás and Castro Díaz, Manuel J. and Parés, Carlos},
title = {Relation between PVM schemes and simple Riemann solvers},
journal = {Numerical Methods for Partial Differential Equations},
year = {2014},
volume = {30},
number = {4},
pages = {1315--1341},
url = {http://onlinelibrary.wiley.com/doi/10.1002/num.21871/abstract},
doi = {10.1002/num.21871}
}
|
| Berthon, C., Foucher, F. and Morales de Luna, T. (2015), "An efficient splitting technique for two-layer shallow-water model", Numerical Methods for Partial Differential Equations., September, 2015. Vol. 31(5), pp. 1396-1423. |
| Abstract: We consider the numerical approximation of the weak solutions of the two-layer shallow-water equations. The model under consideration is made of two usual one-layer shallow-water model coupled by nonconservative products. Because of the nonconservative products of the system, which couple both one-layer shallow-water subsystems, the usual numerical methods have to consider the full model. Of course, uncoupled numerical techniques, just involving finite volume schemes for the basic shallow-water equations, are very attractive since they are very easy to implement and they are costless. Recently, a stable layer splitting technique was introduced [Bouchut and Morales de Luna, M2AN Math Model Numer Anal 42 (2008), 683–698]. In the same spirit, we exhibit new splitting technique, which is proved to be well balanced and non-negative preserving. The main benefit issuing from the here derived uncoupled method is the ability to correctly approximate the solution of very severe benchmarks. |
BibTeX:
@article{berthon2015efficient,
author = {Berthon, Christophe and Foucher, Françoise and Morales de Luna, Tomás},
title = {An efficient splitting technique for two-layer shallow-water model},
journal = {Numerical Methods for Partial Differential Equations},
year = {2015},
volume = {31},
number = {5},
pages = {1396--1423},
url = {http://onlinelibrary.wiley.com/doi/10.1002/num.21949/abstract},
doi = {10.1002/num.21949}
}
|
| Sánchez-Linares, C., Morales de Luna, T. and Castro Díaz, M.J. (2016), "A HLLC scheme for Ripa model", Applied Mathematics and Computation., January, 2016. Vol. 272, Part 2, pp. 369-384. |
| Abstract: 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. |
BibTeX:
@article{sanchez-linares2016hllc,
author = {Sánchez-Linares, C. and Morales de Luna, T. and Castro Díaz, M. J.},
title = {A HLLC scheme for Ripa model},
journal = {Applied Mathematics and Computation},
year = {2016},
volume = {272, Part 2},
pages = {369--384},
url = {http://www.sciencedirect.com/science/article/pii/S0096300315007687},
doi = {10.1016/j.amc.2015.05.137}
}
|
| Morales de Luna, T., Fernández Nieto, E.D. and Castro Díaz, M.J. (2017), "Derivation of a Multilayer Approach to Model Suspended Sediment Transport: Application to Hyperpycnal and Hypopycnal Plumes", Communications in Computational Physics. Vol. 22(5), pp. 1439-1485. |
| Abstract: We propose a multi-layer approach to simulate hyperpycnal and hypopycnal plumes in flows with free surface. The model allows to compute the vertical profile of the horizontal and the vertical components of the velocity of the fluid flow. The model can describe as well the vertical profile of the sediment concentration and the velocity components of each one of the sediment species that form the turbidity current. To do so, it takes into account the settling velocity of the particles and their interaction with the fluid. This allows to better describe the phenomena than a single layer approach. It is in better agreement with the physics of the problem and gives promising results. The numerical simulation is carried out by rewriting the multilayer approach in a compact formulation, which corresponds to a system with nonconservative products, and using path-conservative numerical scheme. Numerical results are presented in order to show the potential of the model. |
BibTeX:
@article{moralesdeluna2017derivation,
author = {Morales de Luna, Tomás and Fernández Nieto, Enrique D.. and Castro Díaz, Manuel J.},
title = {Derivation of a Multilayer Approach to Model Suspended Sediment Transport: Application to Hyperpycnal and Hypopycnal Plumes},
journal = {Communications in Computational Physics},
year = {2017},
volume = {22},
number = {5},
pages = {1439-1485},
doi = {10.4208/cicp.OA-2016-0215}
}
|
| Fernández-Nieto, E.D., Morales de Luna, T., Narbona-Reina, G. and Zabsonré, J.d.D. (2017), "Formal deduction of the Saint-Venant-Exner model including arbitrarily sloping sediment beds and associated energy", ESAIM: Mathematical Modelling and Numerical Analysis., January, 2017. Vol. 51(1), pp. 115-145. |
| Abstract: In this work we present a deduction of the Saint-Venant–Exner model through an asymptotic analysis of the Navier–Stokes equations. A multi-scale analysis is performed in order to take into account that the velocity of the sediment layer is smaller than the one of the fluid layer. This leads us to consider a shallow water type system for the fluid layer and a lubrication Reynolds equation for the sediment one. This deduction provides some improvements with respect to the classic Saint-Venant–Exner model: (i) the deduced model has an associated energy. Moreover, it allows us to explain why classic models do not have an associated energy and how they can be modified in order to recover a model with this property. (ii) The model incorporates naturally a necessary modification that must be taken into account in order to be applied to arbitrarily sloping beds. Furthermore, we show that in general this modification is different from the ones considered classically. Nevertheless, it coincides with a classic one in the case of constant free surface. (iii) The deduced solid transport discharge naturally depends on the thickness of the moving sediment layer, which allows to ensure sediment mass conservation. Moreover, we include a simplified version of the model for the case of quasi-stationary regimes. Some of these simplified models correspond to a generalization of classic ones such as Meyer-Peter and Müller and Ashida–Michiue models. Three numerical tests are presented to study the evolution of a dune for several definition of the repose angle, to see the influence of the proposed definition of the effective shear stress in comparison with the classic one, and by comparing with experimental data. |
BibTeX:
@article{fernandez-nieto2017formal,
author = {Fernández-Nieto, Enrique D. and Morales de Luna, Tomás and Narbona-Reina, Gladys and Zabsonré, Jean de Dieu},
title = {Formal deduction of the Saint-Venant-Exner model including arbitrarily sloping sediment beds and associated energy},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
year = {2017},
volume = {51},
number = {1},
pages = {115--145},
url = {http://dx.doi.org/10.1051/m2an/2016018},
doi = {10.1051/m2an/2016018}
}
|
| Castro, M.J., Morales de Luna, T. and Parés, C. (2017), "Well-Balanced Schemes and Path-Conservative Numerical Methods", In Handbook of Numerical Analysis. Vol. 18, pp. 131-175. Elsevier. |
| Abstract: In this chapter we describe a general methodology for developing high-order well-balanced schemes for hyperbolic system with nonconservative products and/or source terms. We briefly recall the Dal Maso?LeFloch?Murat theory to define weak solutions of nonconservative systems and how it has been used to establish the notion of path-conservative schemes. We show that, under this framework, it is possible to extend to the nonconservative case many well-known numerical schemes that are commonly used for system of conservation laws. Moreover, their extension to high order can be done as well. Next the well-balanced property of the proposed methods is analyzed with an illustrative 1d example. Finally, we point out the difficulties related to the right definition of weak solution and the design of numerical schemes converging to them. |
BibTeX:
@incollection{castro2017well,
author = {Castro, M. J. and Morales de Luna, T. and Parés, C.},
editor = {Shu, Rémi Abgrall and Chi-Wang},
title = {Well-Balanced Schemes and Path-Conservative Numerical Methods},
booktitle = {Handbook of Numerical Analysis},
publisher = {Elsevier},
year = {2017},
volume = {18},
pages = {131--175},
note = {DOI: 10.1016/bs.hna.2016.10.002},
url = {http://www.sciencedirect.com/science/article/pii/S1570865916300333},
doi = {10.1016/bs.hna.2016.10.002}
}
|
| Castro Díaz, M.J., Chalons, C. and Morales de Luna, T. (2018), "A Fully Well-Balanced Lagrange--Projection-Type Scheme for the Shallow-Water Equations", SIAM Journal on Numerical Analysis., jan, 2018. Vol. 56(5), pp. 3071-3098. Society for Industrial & Applied Mathematics (SIAM). |
| Abstract: This work focuses on the numerical approximation of the shallow water equations using a Lagrange--projection-type approach. We propose a fully well-balanced explicit and positive scheme using relevant reconstruction operators. By fully well-balanced, it is meant that the scheme is able to preserve stationary smooth solutions of the model with nonzero velocity, including of course also the well-known lake at rest equilibrium. Numerous numerical experiments illustrate the good behavior of the scheme. Read More: https://epubs.siam.org/doi/10.1137/17M1156101 |
BibTeX:
@article{CastroDiaz2018Fully,
author = {Castro Díaz, Manuel J. and Chalons, Christophe and Morales de Luna, Tomás},
title = {A Fully Well-Balanced Lagrange--Projection-Type Scheme for the Shallow-Water Equations},
journal = {SIAM Journal on Numerical Analysis},
publisher = {Society for Industrial & Applied Mathematics (SIAM)},
year = {2018},
volume = {56},
number = {5},
pages = {3071--3098},
doi = {10.1137/17m1156101}
}
|
| Escalante, C., Morales de Luna, T. and Castro, M.J. (2018), "Non-hydrostatic pressure shallow flows: GPU implementation using finite volume and finite difference scheme", Applied Mathematics and Computation., December, 2018. Vol. 338, pp. 631-659. |
| Abstract: We consider the depth-integrated non-hydrostatic system derived by Yamazaki et al. An efficient formally second-order well-balanced hybrid finite volume finite difference numerical scheme is proposed. The scheme consists of a two-step algorithm based on a projection-correction type scheme initially introduced by Chorin–Temam [15]. First, the hyperbolic part of the system is discretized using a Polynomial Viscosity Matrix path-conservative finite volume method. Second, the dispersive terms are solved by means of compact finite differences. A new methodology is also presented to handle wave breaking over complex bathymetries. This adapts well to GPU-architectures and guidelines about its GPU implementation are introduced. The method has been applied to idealized and challenging experimental test cases, which shows the efficiency and accuracy of the method. |
BibTeX:
@article{Escalante2018Non,
author = {Escalante, C. and Morales de Luna, T. and Castro, M. J.},
title = {Non-hydrostatic pressure shallow flows: GPU implementation using finite volume and finite difference scheme},
journal = {Applied Mathematics and Computation},
year = {2018},
volume = {338},
pages = {631--659},
url = {http://www.sciencedirect.com/science/article/pii/S0096300318305241},
doi = {10.1016/j.amc.2018.06.035}
}
|
| Escalante, C., Fernández-Nieto, E.D., Morales de Luna, T. and Castro, M.J. (2019), "An Efficient Two-Layer Non-hydrostatic Approach for Dispersive Water Waves", Journal of Scientific Computing., April, 2019. Vol. 79(1), pp. 273-320. |
| Abstract: In this paper, we propose a two-layer depth-integrated non-hydrostatic system with improved dispersion relations. This improvement is obtained through three free parameters: two of them related to the representation of the pressure at the interface and a third one that controls the relative position of the interface concerning the total height. These parameters are then optimized to improve the dispersive properties of the resulting system. The optimized model shows good linear wave characteristics up to kH\approx 10𝑘𝐻≈10kH\approx 10, that can be improved for long waves. The system is solved using an efficient formally second-order well-balanced and positive preserving hybrid finite volume/difference numerical scheme. The scheme consists of a two-step algorithm based on a projection-correction type scheme. First, the hyperbolic part of the system is discretized using a Polynomial Viscosity Matrix path-conservative finite-volume method. Second, the dispersive terms are solved using finite differences. The method has been applied to idealized and challenging physical situations that involve nearshore breaking. Agreement with laboratory data is excellent. This technique results in an accurate and efficient method. |
BibTeX:
@article{Escalante2018Efficient,
author = {Escalante, C. and Fernández-Nieto, E. D. and Morales de Luna, T. and Castro, M. J.},
title = {An Efficient Two-Layer Non-hydrostatic Approach for Dispersive Water Waves},
journal = {Journal of Scientific Computing},
year = {2019},
volume = {79},
number = {1},
pages = {273--320},
url = {https://doi.org/10.1007/s10915-018-0849-9},
doi = {10.1007/s10915-018-0849-9}
}
|
| Castro Díaz, M.J., Kurganov, A. and Morales de Luna, T. (2019), "Path-conservative central-upwind schemes for nonconservative hyperbolic systems", ESAIM: Mathematical Modelling and Numerical Analysis. Vol. 53(3), pp. 959-985. |
| Abstract: We develop path-conservative central-upwind schemes for nonconservative one-dimensional hyperbolic systems of nonlinear partial differential equations. Such systems arise in a variety of applications and the most challenging part of their numerical discretization is a robust treatment of nonconservative product terms. Godunov-type central-upwind schemes were developed as an efficient, highly accurate and robust ``black-box’’ solver for hyperbolic systems of conservation and balance laws. They were successfully applied to a large number of hyperbolic systems including several nonconservative ones. To overcome the difficulties related to the presence of nonconservative product terms, several special techniques were proposed. However, none of these techniques was sufficiently robust and thus the applicability of the original central-upwind schemes was rather limited. In this paper, we rewrite the central-upwind schemes in the form of path-conservative schemes. This helps us (i) to show that the main drawback of the original central-upwind approach was the fact that the jump of the nonconservative product terms across cell interfaces has never been taken into account and (ii) to understand how the nonconservative products should be discretized so that their influence on the numerical solution is accurately taken into account. The resulting path-conservative central-upwind scheme is a new robust tool for both conservative and nonconservative hyperbolic systems. We apply the new scheme to the Saint-Venant system with discontinuous bottom topography and two-layer shallow water system. Our numerical results illustrate the good performance of the new path-conservative central-upwind scheme, its robustness and ability to achieve very high resolution. |
BibTeX:
@article{castrodiazmanueljesus2019path,
author = {Castro Díaz, Manuel J. and Kurganov, Alexander and Morales de Luna, Tomás},
title = {Path-conservative central-upwind schemes for nonconservative hyperbolic systems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
year = {2019},
volume = {53},
number = {3},
pages = {959-985},
url = {https://doi.org/10.1051/m2an/2018077},
doi = {10.1051/m2an/2018077}
}
|
| Escalante, C. and Morales de Luna, T. (2020), "A general non-hydrostatic hyperbolic formulation for Boussinesq dispersive shallow flows and its numerical approximation", Journal of Scientific Computing., June, 2020. Vol. 83(3), pp. 62. |
| Abstract: In this paper, we propose a novel first-order reformulation of the most well-known Boussinesq-type systems that are used in ocean engineering. This has the advantage of collecting in a general framework many of the well-known systems used for dispersive flows. Moreover, it avoids the use of high-order derivatives which are not easy to treat numerically, due to the large stencil usually needed. These first-order PDE dispersive systems are then approximated by a novel set of first-order hyperbolic equations. Our new hyperbolic approximation is based on a relaxed augmented system in which the divergence constraints of the velocity flow variables are coupled with the other conservation laws via an evolution equation for the depth-averaged non-hydrostatic pressures. The most important advantage of this new hyperbolic formulation is that it can be easily discretized with explicit and high-order accurate numerical schemes for hyperbolic conservation laws. There is no longer need of solving implicitly some linear system as it is usually done in many classical approaches of Boussinesq-type models. Here a third-order finite volume scheme based on a CWENO reconstruction has been used. The scheme is well-balanced and can treat correctly wet–dry areas and emerging topographies. Several numerical tests, which include idealized academic benchmarks and laboratory experiments are proposed, showing the advantage, efficiency and accuracy of the technique proposed here. |
BibTeX:
@article{escalante2020general,
author = {Escalante, C. and Morales de Luna, T.},
title = {A general non-hydrostatic hyperbolic formulation for Boussinesq dispersive shallow flows and its numerical approximation},
journal = {Journal of Scientific Computing},
year = {2020},
volume = {83},
number = {3},
pages = {62},
url = {https://doi.org/10.1007/s10915-020-01244-7},
doi = {10.1007/s10915-020-01244-7}
}
|
| González-Aguirre, J., Castro, M. and Morales de Luna, T. (2020), "A robust model for rapidly varying flows over movable bottom with suspended and bedload transport: Modelling and numerical approach", Advances in Water Resources., jun, 2020. Vol. 140, pp. 103575. Elsevier BV. |
| Abstract: We propose a coupled model for suspended and bedload sediment transport in the shallow water framework. The model is deduced under hydrostatic pressure assumptions and will not assume any Boussinesq hypothesis. The numerical resolution is carried out in a segregated way. First the underlying system of conservation laws is solved by using a first order path-conservative Riemann solver. Then, the source terms corresponding with the erosion and depositions rates are approximated in a semi-implicit way. The final scheme preserves the positivity of the density. Several numerical experiments were carried out in order to validate the model and the numerical scheme. The results obtained are in good agreement with the experimental data. |
BibTeX:
@article{gonzalezaguirre2020robust,
author = {González-Aguirre, J.C. and Castro, M.J. and Morales de Luna, T.},
title = {A robust model for rapidly varying flows over movable bottom with suspended and bedload transport: Modelling and numerical approach},
journal = {Advances in Water Resources},
publisher = {Elsevier BV},
year = {2020},
volume = {140},
pages = {103575},
doi = {10.1016/j.advwatres.2020.103575}
}
|
| Guerrero Fernández, E., Castro-Díaz, M.J. and Morales de Luna, T. (2020), "A Second-Order Well-Balanced Finite Volume Scheme for the Multilayer Shallow Water Model with Variable Density", Mathematics., May, 2020. Vol. 8(5), pp. 848. |
| Abstract: In this work, we consider a multilayer shallow water model with variable density. It consists of a system of hyperbolic equations with non-conservative products that takes into account the pressure variations due to density fluctuations in a stratified fluid. A second-order finite volume method that combines a hydrostatic reconstruction technique with a MUSCL second order reconstruction operator is developed. The scheme is well-balanced for the lake-at-rest steady state solutions. Additionally, hints on how to preserve a general class of stationary solutions corresponding to a stratified density profile are also provided. Some numerical results are presented, including validation with laboratory data that show the efficiency and accuracy of the approach introduced here. Finally, a comparison between two different parallelization strategies on GPU is presented. |
BibTeX:
@article{guerrerofernandez2020second,
author = {Guerrero Fernández, Ernesto and Castro-Díaz, Manuel Jesús and Morales de Luna, Tomás},
title = {A Second-Order Well-Balanced Finite Volume Scheme for the Multilayer Shallow Water Model with Variable Density},
journal = {Mathematics},
year = {2020},
volume = {8},
number = {5},
pages = {848},
url = {https://www.mdpi.com/2227-7390/8/5/848},
doi = {10.3390/math8050848}
}
|
| Morales de Luna, T., Castro Díaz, M.J. and Chalons, C. (2020), "High-order fully well-balanced lagrange-projection scheme for shallow water", Communications in Mathematical Sciences. Vol. 18(3), pp. 781-807. |
| Abstract: In this work we propose a novel strategy to define high-order fully well-balanced Lagrange-projection finite volume solvers for balance laws. In particular, we focus on the 1D shallow water system as it is a reference system of balance laws with non-trivial stationary solutions. Nevertheless, the strategy proposed here could be extended to other interesting balance laws. By fully well-balanced, it is meant that the scheme is able to preserve stationary smooth solutions. Following [M.J. Castro et al., in Handbook of Numerical Analysis, 18:131175, 2017], we exploit the idea of using a high-order well-balanced reconstruction operator for the Lagrangian step. Nevertheless, this is not enough to achieve well-balanced high-order during the projection step. We propose here a new projection step that overcomes this difficulty and that reduces to the standard one in case of conservation laws. Finally, some numerical experiments illustrate the good behaviour of the scheme. |
BibTeX:
@article{castrodiaz2020high,
author = {Morales de Luna, Tomás and Castro Díaz, Manuel J. and Chalons, Christophe},
title = {High-order fully well-balanced lagrange-projection scheme for shallow water},
journal = {Communications in Mathematical Sciences},
year = {2020},
volume = {18},
number = {3},
pages = {781--807},
doi = {10.4310/CMS.2020.v18.n3.a9}
}
|
| Garres-Díaz, J., Fernández-Nieto, E.D., Mangeney, A. and Morales de Luna, T. (2021), "A Weakly Non-hydrostatic Shallow Model for Dry Granular Flows", Journal of Scientific Computing., jan, 2021. Vol. 86(2) Springer Science and Business Media LLC. |
| Abstract: A non-hydrostatic depth-averaged model for dry granular flows is proposed, taking into account vertical acceleration. A variable friction coefficient based on the $$\mu (I)$$rheology is considered. The model is obtained from an asymptotic analysis in a local reference system, where the non-hydrostatic contribution is supposed to be small compared to the hydrostatic one. The non-hydrostatic counterpart of the pressure may be written as the sum of two terms: one corresponding to the stress tensor and the other to the vertical acceleration. The model introduced here is weakly non-hydrostatic, in the sense that the non-hydrostatic contribution related to the stress tensor is not taken into account due to its complex implementation. The motivation is to propose simple models including non-hydrostatic effects. In order to approximate the resulting model, a simple and efficient numerical scheme is proposed. It consists of a three-step splitting procedure and the resulting scheme is well-balanced for granular material at rest with slope smaller than the fixed repose angle. The model and numerical scheme are validated by means of several numerical tests, including a convergence test, a well-balanced test, and comparisons with laboratory experiments of granular collapse. The influence of non-hydrostatic terms and of the choice of the coordinate system (Cartesian or local) is also analyzed. We show that non-hydrostatic models are less sensitive to the choice of the coordinate system. In addition, the non-hydrostatic Cartesian model produces deposits similar to the hydrostatic local model as suggested by Denlinger and Iverson (J Geophys Res Earth Surf, 2004. https://doi.org/10.1029/2003jf000085), the flow dynamics being however different. Moreover, the proposed model, when written in Cartesian coordinates, can be seen as an improvement of their model, since the vertical velocity is computed and not estimated from the boundary conditions. In general, the non-hydrostatic model introduced here much better reproduces granular collapse experiments compared to hydrostatic models, especially at the beginning of the flow. |
BibTeX:
@article{garresdiaz2021weakly,
author = {Garres-Díaz, J. and Fernández-Nieto, E. D. and Mangeney, A. and Morales de Luna, T.},
title = {A Weakly Non-hydrostatic Shallow Model for Dry Granular Flows},
journal = {Journal of Scientific Computing},
publisher = {Springer Science and Business Media LLC},
year = {2021},
volume = {86},
number = {2},
doi = {10.1007/s10915-020-01377-9}
}
|
| Escalante, C., Fernández-Nieto, E.D., Morales de Luna, T., Penel, Y. and Sainte-Marie, J. (2021), "Numerical Simulations of a Dispersive Model Approximating Free-Surface Euler Equations", Journal of Scientific Computing., oct, 2021. Vol. 89(3) Springer Science and Business Media LLC. |
BibTeX:
@article{escalante2021numericala,
author = {Escalante, Cipriano and Fernández-Nieto, Enrique D. and Morales de Luna, Tomás and Penel, Yohan and Sainte-Marie, Jacques},
title = {Numerical Simulations of a Dispersive Model Approximating Free-Surface Euler Equations},
journal = {Journal of Scientific Computing},
publisher = {Springer Science and Business Media LLC},
year = {2021},
volume = {89},
number = {3},
doi = {10.1007/s10915-021-01552-6}
}
|
| Garres-Díaz, J., Castro Díaz, M.J., Koellermeier, J. and Morales de Luna, T. (2021), "Shallow Water Moment Models for Bedload Transport Problems", Communications in Computational Physics., jun, 2021. Vol. 30(3), pp. 903-941. Global Science Press. |
| Abstract: In this work a simple but accurate shallow model for bedload sediment transport is proposed. The model is based on applying the moment approach to the Shallow Water Exner model, making it possible to recover the vertical structure of the flow. This approach allows us to obtain a better approximation of the fluid velocity close to the bottom, which is the relevant velocity for the sediment transport. A general Shallow Water Exner moment model allowing for polynomial velocity profiles of arbitrary order is obtained. A regularization ensures hyperbolicity and easy computation of the eigenvalues. The system is solved by means of an adapted IFCP scheme proposed here. The improvement of this IFCP type scheme is based on the approximation of the eigenvalue associated to the sediment transport. Numerical tests are presented which deal with large and short time scales. The proposed model allows to obtain the vertical structure of the fluid, which results in a better description on the bedload transport of the sediment layer. |
BibTeX:
@article{garresdiaz2021shallow,
author = {Garres-Díaz, J. and Castro Díaz, M. J. and Koellermeier, J. and Morales de Luna, T.},
title = {Shallow Water Moment Models for Bedload Transport Problems},
journal = {Communications in Computational Physics},
publisher = {Global Science Press},
year = {2021},
volume = {30},
number = {3},
pages = {903--941},
doi = {10.4208/cicp.oa-2020-0152}
}
|
| Guerrero Fernández, E., Castro Díaz, M.J., Dumbser, M. and Morales de Luna, T. (2022), "An Arbitrary High Order Well-Balanced ADER-DG Numerical Scheme for the Multilayer Shallow-Water Model with Variable Density", Journal of Scientific Computing., dec, 2022. Vol. 90(52) Springer Science and Business Media LLC. |
BibTeX:
@article{guerrerofernandez2022arbitrary,
author = {Guerrero Fernández, E. and Castro Díaz,M. J. and Dumbser,M. and Morales de Luna,T.},
title = {An Arbitrary High Order Well-Balanced ADER-DG Numerical Scheme for the Multilayer Shallow-Water Model with Variable Density},
journal = {Journal of Scientific Computing},
publisher = {Springer Science and Business Media LLC},
year = {2022},
volume = {90},
number = {52},
doi = {10.1007/s10915-021-01734-2}
}
|
| Pimentel-García, E., Castro, M.J., Chalons, C., Morales de Luna, T. and Parés, C. (2022), "In-cell Discontinuous Reconstruction path-conservative methods for non conservative hyperbolic systems - Second-order extension", Journal of Computational Physics., mar, 2022. , pp. 111152. Elsevier BV. |
BibTeX:
@article{pimentelgarcia2022cell,
author = {Pimentel-García, E. and Castro, Manuel J. and Chalons, C. and Morales de Luna,T. and Parés, C.},
title = {In-cell Discontinuous Reconstruction path-conservative methods for non conservative hyperbolic systems - Second-order extension},
journal = {Journal of Computational Physics},
publisher = {Elsevier BV},
year = {2022},
pages = {111152},
doi = {10.1016/j.jcp.2022.111152}
}
|
| Garres-Díaz, J., Escalante, C., Morales de Luna, T. and Castro, M.J. (2023), "A general vertical decomposition of Euler equations: Multilayer-moment models", Applied Numerical Mathematics., jan, 2023. Vol. 183, pp. 236-262. Elsevier BV. |
BibTeX:
@article{garresdiaz2023general,
author = {Garres-Díaz, J. and Escalante, C. and Morales de Luna, T. and Castro, M. J.},
title = {A general vertical decomposition of Euler equations: Multilayer-moment models},
journal = {Applied Numerical Mathematics},
publisher = {Elsevier BV},
year = {2023},
volume = {183},
pages = {236--262},
doi = {10.1016/j.apnum.2022.09.004}
}
|
| Caballero-Cárdenas, C., Castro, M., Morales de Luna, T. and Muñoz-Ruiz, M. (2023), "Implicit and implicit-explicit Lagrange-projection finite volume schemes exactly well-balanced for 1D shallow water system", Applied Mathematics and Computation., apr, 2023. Vol. 443, pp. 127784. Elsevier BV. |
BibTeX:
@article{caballerocardenas2023implicit,
author = {Caballero-Cárdenas, C. and Castro, M.J. and Morales de Luna, T. and Muñoz-Ruiz, M.L.},
title = {Implicit and implicit-explicit Lagrange-projection finite volume schemes exactly well-balanced for 1D shallow water system},
journal = {Applied Mathematics and Computation},
publisher = {Elsevier BV},
year = {2023},
volume = {443},
pages = {127784},
doi = {10.1016/j.amc.2022.127784}
}
|
| Escalante, C., Fernández-Nieto, E., Garres-Díaz, J., Penel, Y. and Morales de Luna, T. (2023), "Non-hydrostatic layer-averaged approximation of Euler system with enhanced dispersion properties", Computational & Applied Mathematics. Springer Science and Business Media LLC. |
| Abstract: A new family of non-hydrostatic layer-averaged models for the non-stationary Euler equations is presented in this work, with improved dispersion relations. They are a generalisation of the layer-averaged models introduced in [12], named LDNH models, where the vertical profile of the horizontal velocity is layerwise constant. This assumption implies that solutions of LDNH can be seen as a first order Galerkin approximation of Euler system. Nevertheless, it is not a fully (x, z) Galerkin discretisation of Euler system, but just in the vertical direction (z). Thus, the resulting model only depends on the horizontal space variable (x), and therefore specific and efficient numerical methods can be applied (see [9]). This work focuses on particular weak solutions where the horizontal velocity is layerwise linear on z and possibly discontinuous across layer interfaces. This approach allows the system to be a second-order approximation in the vertical direction of Euler system. Several closure relations of the layer-averaged system with non-hydrostatic pressure are presented. The resulting models are named LIN-NHk models, with k = 0, 1, 2. Parameter k indicates the degree of the vertical velocity profile considered in the approximation of the vertical momentum equation. All the introduced models satisfy a dissipative energy balance. Finally, an analysis and a comparison of the dispersive properties of each model are carried out. We show that Models LIN-NH1 and LIN-NH2 provide a better dispersion relation, group velocity and shoaling than LDNH models. |
BibTeX:
@article{escalante2023non,
author = {Escalante, C. and Fernández-Nieto, E.D. and Garres-Díaz, J. and Penel, Y. and Morales de Luna, T.},
title = {Non-hydrostatic layer-averaged approximation of Euler system with enhanced dispersion properties},
journal = {Computational & Applied Mathematics},
publisher = {Springer Science and Business Media LLC},
year = {2023}
}
|
| Del Grosso, A., Castro, M., Chalons, C. and Morales de Luna, T. (2023), "On well-balanced implicit-explicit Lagrange-projection schemes for two-layer shallow water equations", Applied Mathematics and Computation., apr, 2023. Vol. 442, pp. 127702. Elsevier BV. |
BibTeX:
@article{grosso2023well,
author = {Del Grosso, A. and Castro, M. and Chalons, C. and Morales de Luna, T.},
title = {On well-balanced implicit-explicit Lagrange-projection schemes for two-layer shallow water equations},
journal = {Applied Mathematics and Computation},
publisher = {Elsevier BV},
year = {2023},
volume = {442},
pages = {127702},
doi = {10.1016/j.amc.2022.127702}
}
|