María Luz Muñoz-Ruiz | Universidad de Málaga (original) (raw)
Papers by María Luz Muñoz-Ruiz
Differential and Integral Equations, 2004
In this paper we prove the existence of a solution for a nonhomogeneous bi-layer shallow-water mo... more In this paper we prove the existence of a solution for a nonhomogeneous bi-layer shallow-water model in depth-mean velocity formulation. In [7] the homogeneous case was studied. The main difficulties in the nonhomogeneous case arise from the treatment of the boundary terms.
Applied Mathematics and Computation
Nonlinear Analysis-real World Applications, Mar 1, 2003
In this paper, we prove an existence and uniqueness result for a bi-layer shallow water model in ... more In this paper, we prove an existence and uniqueness result for a bi-layer shallow water model in depth-mean velocity formulation. Some smoothness results for the solution are also obtained. In a previous work we proved the same results for a one-layer problem. Now the di culty arises from the terms coupling the two layers. In order to obtain the energy estimate, we use a special basis which allows us to bound these terms.
XVII Congreso de Ecuaciones Diferenciales y Aplicaciones, VII Congreso de Matemática Aplicada: Salamanca, 14-28 septiembre 2001, 2001, ISBN 8469961446, págs. 653-654, 2001
journal homepage: www.elsevier.com/locate/jcp Why many theories of shock waves are necessary: Con... more journal homepage: www.elsevier.com/locate/jcp Why many theories of shock waves are necessary: Convergence error
El objetivo principal de esta memoria es el analisis teorico de un sistema de ecuaciones en deriv... more El objetivo principal de esta memoria es el analisis teorico de un sistema de ecuaciones en derivadas parciales acopladas, que corresponde a un modelo bicapa de aguas poco profundas, aplicable al estudio de la dinamica que tiene lugar en la zona del Mar de Alboran y el Estrecho de Gibraltar, En primer lugar se aborda la construccion del modelo de aguas poco profundas bicapa en formulacion velocidad-espesor, y se recuerdan aquellos resultados relativos al analisis de un modelo de aguas poco profundas de una sola capa que se extenderan al caso de dos capas. A continuacion se aborda el analisis del problema bicapa con condiciones de contorno homogeneas, ofreciendo un teorema de existencia de solucion para datos controlados, y algunos resultados de regularidad que permiten probar un teorema de unicidad de solucion. La principal dificultad es la aparicion de terminos de acoplamiento entre las capas. Esto plantea la necesidad de obtener estimaciones a priori en el espacio de las funciones de cuadrado sumable con el fin de obtener la existencia de soluciones. En el caso de una sola capa estas estimaciones eran obtenidas una vez probada la existencia de solucion. Despues se estudia el problema bicapa con condiciones de contorno no homogeneas, para el que se ofrece un teorema de exitencia de solucion. A las diferencias propias del modelo bicapa se une ahora las debidas a la aparicion de terminos de borde que es necesario estimar. Por ultimo se realiza el analisis teorico y la aproximacion numerica de un problema bicapa unidimensional que modela un flujo bicapa en un canal con seccion rectangular variable. Desde el punto de vista teorico se presentan resultados de existencia, regularidad y unicidad de solucion. Desde el punto de vista numerico se propone un esquema de tipo volumenes finitos para su resolucion: El Q-esquema de Van Leer con descentrado de los terminos fuente y se presentan algunos resultados nu
Numerical Methods for Hyperbolic Equations, 2012
Actas del Encuentro de …, 2001
Informaci??n del art??culo Sobre un problema Shallow-Water bi-capas: resultados de existencia y u... more Informaci??n del art??culo Sobre un problema Shallow-Water bi-capas: resultados de existencia y unicidad.
congreso.us.es
En este trabajo se aborda la aproximación numérica del problema de Cauchy para sistemas hiperbóli... more En este trabajo se aborda la aproximación numérica del problema de Cauchy para sistemas hiperbólicos no conservativos en dimensión uno. Para definir el concepto de solución débil de di-chos sistemas utilizamos la teorıa desarrollada por Dal Maso, Le Floch y Murat, según ...
Con: Matematicas del planeta Tierra : [cuaderno de actividades] / Fernando Alcaide, Miguel Nieto
Numerical Mathematics and Advanced Applications, 2008
ABSTRACT We are concerned with the numerical approximation of Cauchy problems for hyperbolic syst... more ABSTRACT 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 Parés) 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.
Nonlinear Analysis: Theory, Methods & Applications, 2003
This paper is concerned with the mathematical analysis and the numerical approximation of the sys... more This paper is concerned with the mathematical analysis and the numerical approximation of the system of partial di erential equations governing the one-dimensional ow of two superposed shallow layers of immiscible viscous uid in a channel with variable rectangular cross-section. First, we prove the existence and uniqueness of solution for small data and some smoothness results. Next, a ÿrst-order upwind scheme for numerically solving the system is proposed. We apply this scheme to the simulation of some two-layer exchange ows through straits with a sill and a contraction.
Theory, Numerics and Applications (In 2 Volumes), 2012
Proceedings of Symposia in Applied Mathematics, 2009
Computational Fluid and Solid Mechanics, 2001
Nonlinear Analysis, 2004
In this paper we prove the existence of a solution for a nonhomogeneous bi-layer shallow-water mo... more In this paper we prove the existence of a solution for a nonhomogeneous bi-layer shallow-water model in depth-mean velocity formulation. In [7] the homogeneous case was studied. The main difficulties in the nonhomogeneous case arise from the treatment of the boundary terms.
Journal of Scientific Computing, 2011
This paper deals with the numerical approximation of one-dimensional hyperbolic systems of balanc... more This paper deals with the numerical approximation of one-dimensional hyperbolic systems of balance laws. We consider these systems as a particular case of hyperbolic systems in nonconservative form, for which we use the theory introduced by Dal Maso, LeFloch and Murat (J. Math. Pures Appl. 74:483, 1995) in order to define the concept of weak solutions. This theory is based
Journal of Computational Physics, 2008
We are interested in nonlinear hyperbolic systems in nonconservative form arising in fluid dynami... more We are interested in nonlinear hyperbolic systems in nonconservative form arising in fluid dynamics, and, for solutions containing shock waves, we investigate the convergence of finite difference schemes applied to such systems. According to Dal Maso, LeFloch, and Murat's theory, a shock wave theory for a given nonconservative system requires prescribing a priori a family of paths in the phase space. In the present paper, we consider schemes that are formally consistent with a given family of paths, and we investigate their limiting behavior as the mesh is refined. we first generalize to systems a property established earlier by Hou and LeFloch for scalar conservation laws, and we prove that nonconservative schemes generate, at the level of the limiting hyperbolic system, an convergence error source-term which, provided the total variation of the approximations remains uniformly bounded, is a locally bounded measure. This convergence error measure is supported on the shock trajectories and, as we demonstrate here, is usually "small". In the special case that the scheme converges in the sense of graphs-a rather strong convergence property often violated in practice-then this measure source-term vanishes. We also discuss the role of the equivalent equation associated with a difference scheme; here, the distinction between scalar equations and systems appears most clearly since, for systems, the equivalent equation of a scheme that is formally path-consistent depends upon the prescribed family of paths. The core of this paper is devoted to investigate numerically the approximation of several (simplified or full) hyperbolic models arising in fluid dynamics. This leads us to the
ESAIM: Mathematical Modelling and Numerical Analysis, 2007
This paper is concerned with the numerical approximation of Cauchy problems for onedimensional no... more This paper is concerned with the numerical approximation of Cauchy problems for onedimensional nonconservative hyperbolic systems. The theory developed by Dal Maso et al. [J. Math. Pures Appl. 74 (1995) 483-548] is used in order to define the weak solutions of the system: an interpretation of the nonconservative products as Borel measures is given, based on the choice of a family of paths drawn in the phase space. Even if the family of paths can be chosen arbitrarily, it is natural to require this family to satisfy some hypotheses concerning the relation of the paths with the integral curves of the characteristic fields. The first goal of this paper is to investigate the implications of three basic hypotheses of this nature. Next, we show that, when the family of paths satisfies these hypotheses, Godunov methods can be written in a natural form that generalizes their classical expression for systems of conservation laws. We also study the well-balance properties of these methods. Finally, we prove the consistency of the numerical scheme with the definition of weak solutions: we prove that, under hypothesis of bounded total variation, if the approximations provided by a Godunov method based on a family of paths converge uniformly to some function as the mesh is refined, then this function is a weak solution (related to that family of paths) of the nonconservative system. We extend this result to a family of numerical schemes based on approximate Riemann solvers.
Bol. Soc. Esp. Mat. Apl, 2009
... Apl. no47(2009), 1948 ON SOME DIFFICULTIES OF THE NUMERICAL APPROXIMATION OF NONCONSERVATIVE... more ... Apl. no47(2009), 1948 ON SOME DIFFICULTIES OF THE NUMERICAL APPROXIMATION OF NONCONSERVATIVE HYPERBOLIC SYSTEMS CARLOS PARÉS, MARıA LUZ MUNOZ-RUIZ Departamento de Análisis Matemático Universidad de Málaga, 29071 Málaga, Spain ...
Differential and Integral Equations, 2004
In this paper we prove the existence of a solution for a nonhomogeneous bi-layer shallow-water mo... more In this paper we prove the existence of a solution for a nonhomogeneous bi-layer shallow-water model in depth-mean velocity formulation. In [7] the homogeneous case was studied. The main difficulties in the nonhomogeneous case arise from the treatment of the boundary terms.
Applied Mathematics and Computation
Nonlinear Analysis-real World Applications, Mar 1, 2003
In this paper, we prove an existence and uniqueness result for a bi-layer shallow water model in ... more In this paper, we prove an existence and uniqueness result for a bi-layer shallow water model in depth-mean velocity formulation. Some smoothness results for the solution are also obtained. In a previous work we proved the same results for a one-layer problem. Now the di culty arises from the terms coupling the two layers. In order to obtain the energy estimate, we use a special basis which allows us to bound these terms.
XVII Congreso de Ecuaciones Diferenciales y Aplicaciones, VII Congreso de Matemática Aplicada: Salamanca, 14-28 septiembre 2001, 2001, ISBN 8469961446, págs. 653-654, 2001
journal homepage: www.elsevier.com/locate/jcp Why many theories of shock waves are necessary: Con... more journal homepage: www.elsevier.com/locate/jcp Why many theories of shock waves are necessary: Convergence error
El objetivo principal de esta memoria es el analisis teorico de un sistema de ecuaciones en deriv... more El objetivo principal de esta memoria es el analisis teorico de un sistema de ecuaciones en derivadas parciales acopladas, que corresponde a un modelo bicapa de aguas poco profundas, aplicable al estudio de la dinamica que tiene lugar en la zona del Mar de Alboran y el Estrecho de Gibraltar, En primer lugar se aborda la construccion del modelo de aguas poco profundas bicapa en formulacion velocidad-espesor, y se recuerdan aquellos resultados relativos al analisis de un modelo de aguas poco profundas de una sola capa que se extenderan al caso de dos capas. A continuacion se aborda el analisis del problema bicapa con condiciones de contorno homogeneas, ofreciendo un teorema de existencia de solucion para datos controlados, y algunos resultados de regularidad que permiten probar un teorema de unicidad de solucion. La principal dificultad es la aparicion de terminos de acoplamiento entre las capas. Esto plantea la necesidad de obtener estimaciones a priori en el espacio de las funciones de cuadrado sumable con el fin de obtener la existencia de soluciones. En el caso de una sola capa estas estimaciones eran obtenidas una vez probada la existencia de solucion. Despues se estudia el problema bicapa con condiciones de contorno no homogeneas, para el que se ofrece un teorema de exitencia de solucion. A las diferencias propias del modelo bicapa se une ahora las debidas a la aparicion de terminos de borde que es necesario estimar. Por ultimo se realiza el analisis teorico y la aproximacion numerica de un problema bicapa unidimensional que modela un flujo bicapa en un canal con seccion rectangular variable. Desde el punto de vista teorico se presentan resultados de existencia, regularidad y unicidad de solucion. Desde el punto de vista numerico se propone un esquema de tipo volumenes finitos para su resolucion: El Q-esquema de Van Leer con descentrado de los terminos fuente y se presentan algunos resultados nu
Numerical Methods for Hyperbolic Equations, 2012
Actas del Encuentro de …, 2001
Informaci??n del art??culo Sobre un problema Shallow-Water bi-capas: resultados de existencia y u... more Informaci??n del art??culo Sobre un problema Shallow-Water bi-capas: resultados de existencia y unicidad.
congreso.us.es
En este trabajo se aborda la aproximación numérica del problema de Cauchy para sistemas hiperbóli... more En este trabajo se aborda la aproximación numérica del problema de Cauchy para sistemas hiperbólicos no conservativos en dimensión uno. Para definir el concepto de solución débil de di-chos sistemas utilizamos la teorıa desarrollada por Dal Maso, Le Floch y Murat, según ...
Con: Matematicas del planeta Tierra : [cuaderno de actividades] / Fernando Alcaide, Miguel Nieto
Numerical Mathematics and Advanced Applications, 2008
ABSTRACT We are concerned with the numerical approximation of Cauchy problems for hyperbolic syst... more ABSTRACT 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 Parés) 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.
Nonlinear Analysis: Theory, Methods & Applications, 2003
This paper is concerned with the mathematical analysis and the numerical approximation of the sys... more This paper is concerned with the mathematical analysis and the numerical approximation of the system of partial di erential equations governing the one-dimensional ow of two superposed shallow layers of immiscible viscous uid in a channel with variable rectangular cross-section. First, we prove the existence and uniqueness of solution for small data and some smoothness results. Next, a ÿrst-order upwind scheme for numerically solving the system is proposed. We apply this scheme to the simulation of some two-layer exchange ows through straits with a sill and a contraction.
Theory, Numerics and Applications (In 2 Volumes), 2012
Proceedings of Symposia in Applied Mathematics, 2009
Computational Fluid and Solid Mechanics, 2001
Nonlinear Analysis, 2004
In this paper we prove the existence of a solution for a nonhomogeneous bi-layer shallow-water mo... more In this paper we prove the existence of a solution for a nonhomogeneous bi-layer shallow-water model in depth-mean velocity formulation. In [7] the homogeneous case was studied. The main difficulties in the nonhomogeneous case arise from the treatment of the boundary terms.
Journal of Scientific Computing, 2011
This paper deals with the numerical approximation of one-dimensional hyperbolic systems of balanc... more This paper deals with the numerical approximation of one-dimensional hyperbolic systems of balance laws. We consider these systems as a particular case of hyperbolic systems in nonconservative form, for which we use the theory introduced by Dal Maso, LeFloch and Murat (J. Math. Pures Appl. 74:483, 1995) in order to define the concept of weak solutions. This theory is based
Journal of Computational Physics, 2008
We are interested in nonlinear hyperbolic systems in nonconservative form arising in fluid dynami... more We are interested in nonlinear hyperbolic systems in nonconservative form arising in fluid dynamics, and, for solutions containing shock waves, we investigate the convergence of finite difference schemes applied to such systems. According to Dal Maso, LeFloch, and Murat's theory, a shock wave theory for a given nonconservative system requires prescribing a priori a family of paths in the phase space. In the present paper, we consider schemes that are formally consistent with a given family of paths, and we investigate their limiting behavior as the mesh is refined. we first generalize to systems a property established earlier by Hou and LeFloch for scalar conservation laws, and we prove that nonconservative schemes generate, at the level of the limiting hyperbolic system, an convergence error source-term which, provided the total variation of the approximations remains uniformly bounded, is a locally bounded measure. This convergence error measure is supported on the shock trajectories and, as we demonstrate here, is usually "small". In the special case that the scheme converges in the sense of graphs-a rather strong convergence property often violated in practice-then this measure source-term vanishes. We also discuss the role of the equivalent equation associated with a difference scheme; here, the distinction between scalar equations and systems appears most clearly since, for systems, the equivalent equation of a scheme that is formally path-consistent depends upon the prescribed family of paths. The core of this paper is devoted to investigate numerically the approximation of several (simplified or full) hyperbolic models arising in fluid dynamics. This leads us to the
ESAIM: Mathematical Modelling and Numerical Analysis, 2007
This paper is concerned with the numerical approximation of Cauchy problems for onedimensional no... more This paper is concerned with the numerical approximation of Cauchy problems for onedimensional nonconservative hyperbolic systems. The theory developed by Dal Maso et al. [J. Math. Pures Appl. 74 (1995) 483-548] is used in order to define the weak solutions of the system: an interpretation of the nonconservative products as Borel measures is given, based on the choice of a family of paths drawn in the phase space. Even if the family of paths can be chosen arbitrarily, it is natural to require this family to satisfy some hypotheses concerning the relation of the paths with the integral curves of the characteristic fields. The first goal of this paper is to investigate the implications of three basic hypotheses of this nature. Next, we show that, when the family of paths satisfies these hypotheses, Godunov methods can be written in a natural form that generalizes their classical expression for systems of conservation laws. We also study the well-balance properties of these methods. Finally, we prove the consistency of the numerical scheme with the definition of weak solutions: we prove that, under hypothesis of bounded total variation, if the approximations provided by a Godunov method based on a family of paths converge uniformly to some function as the mesh is refined, then this function is a weak solution (related to that family of paths) of the nonconservative system. We extend this result to a family of numerical schemes based on approximate Riemann solvers.
Bol. Soc. Esp. Mat. Apl, 2009
... Apl. no47(2009), 1948 ON SOME DIFFICULTIES OF THE NUMERICAL APPROXIMATION OF NONCONSERVATIVE... more ... Apl. no47(2009), 1948 ON SOME DIFFICULTIES OF THE NUMERICAL APPROXIMATION OF NONCONSERVATIVE HYPERBOLIC SYSTEMS CARLOS PARÉS, MARıA LUZ MUNOZ-RUIZ Departamento de Análisis Matemático Universidad de Málaga, 29071 Málaga, Spain ...