Pep Mulet
Universitat de València, Matematica Aplicada, Faculty Member
... to traditional methods and naturally a part of the research in this area is directed towards making them ... ear problem, as well as for the linear algebra problems which arise at each step. ... For simplicity, we will consider the... more
... to traditional methods and naturally a part of the research in this area is directed towards making them ... ear problem, as well as for the linear algebra problems which arise at each step. ... For simplicity, we will consider the unconstrained formulation of the Tikhonov regularization. ...
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Polydisperse suspensions processes are usually described by one-dimensional sys- tems of conservation laws. The hyperbolicity problem of these systems is an important property, because it is often related to the range of validity of the... more
Polydisperse suspensions processes are usually described by one-dimensional sys- tems of conservation laws. The hyperbolicity problem of these systems is an important property, because it is often related to the range of validity of the models. For some models, the particu- lar structure of the Jacobian of ux function is exploited to give a proof of hyperbolicity. In the simulations we use a improved version of the classical WENO scheme, namely, mapped WENO (4), along with a fourth order and four steps optimal strong-stability preserving Runge-Kutta time discretization given by Gottlieb and Ruuth (3). These authors developed an algorithm for the popular fth order WENO (WENO5) that lets us compute downwind operator values with a low computational cost. We incorporate the mapped version to this algorithm.
Research Interests: Mathematics, Image Processing, Iterative Methods, TV, Numerical Methods, and 13 moreNumerical Method, Color, Deconvolution, Image Restoration, Convolution, Color Image, Interpolation, Blind Deconvolution, Euler Lagrange Equation, Preconditioning, Nonlinear Equations, Color images, and Total Variation
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Summary. We review a conservative finite difference shock capturing scheme that has been used by our research team over the last years for the numerical simula-tions of complex flows [3, 6]. This scheme is based on Shu and... more
Summary. We review a conservative finite difference shock capturing scheme that has been used by our research team over the last years for the numerical simula-tions of complex flows [3, 6]. This scheme is based on Shu and Osher's technique [9] for the design of highly ...
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ABSTRACT Multiclass Lighthill-Whitham-Richards traffic models [S. Benzoni-Gavage and R. M. Colombo, Eur. J. Appl. Math. 14, No. 5, 587–612 (2003; Zbl 1143.82323; G. C. K. Wong and S. C. Wong, “A multi-class traffic flow model – an... more
ABSTRACT Multiclass Lighthill-Whitham-Richards traffic models [S. Benzoni-Gavage and R. M. Colombo, Eur. J. Appl. Math. 14, No. 5, 587–612 (2003; Zbl 1143.82323; G. C. K. Wong and S. C. Wong, “A multi-class traffic flow model – an extension of LWR model with heterogeneous drivers”, Transp. Res. Part A 36, 827–841 (2002)] give rise to first-order systems of conservation laws that are hyperbolic under usual conditions, so that their associated Cauchy problems are wellposed. Anticipation lengths and reaction times can be incorporated into these models by adding certain conservative second-order terms to these first-order conservation laws. These terms can be diffusive under certain circumstances, thus, in principle, ensuring the stability of the solutions. The purpose of this paper is to analyze the stability of these diffusively corrected models under varying reaction times and anticipation lengths. It is demonstrated that instabilities may develop for high reaction times and short anticipation lengths, and that these instabilities may have controlled frequencies and amplitudes due to their nonlinear nature.
ABSTRACT In this paper we provide a derivation of a 1D N-phase flow model in a porous medium under the condition of vertical equilibrium which generalizes the three-phase flow model developed in Guerrero et al. (2013). We identify, from a... more
ABSTRACT In this paper we provide a derivation of a 1D N-phase flow model in a porous medium under the condition of vertical equilibrium which generalizes the three-phase flow model developed in Guerrero et al. (2013). We identify, from a mathematical point of view, a set of conditions on the capillary pressures that ensure that the linearized, purely parabolic, initial value problem is well-posed. For the numerical simulation of the model, we advocate the use of Implicit–Explicit (IMEX)–Runge–Kutta (RK) schemes for time evolution with a Weighted-Essentially Non-Oscillatory (WENO) spatial discretization of the convective terms. In previous papers (Donat et al., 2013), (Guerrero et al., 2013), we showed that IMEX–RK methods can be useful in the numerical simulation of 1D two-phase and three-phase flows, since the stability restrictions for the time-step of these schemes are less severe than those of fully explicit schemes. On the other hand, their implementation requires, in general, a fairly intensive use of a nonlinear system solver, so that the efficiency and robustness of IMEX schemes for multi-phase flow is directly related to this nonlinear technique. In this paper we describe an efficient nonlinear system solver, based on an appropriate fixed-point iteration technique, in order to find the solution of the nonlinear systems that result from the implicit discretization of the nonlinear diffusive terms in the model. In addition, we implement zero-flux boundary conditions fully consistent with the vertical equilibrium assumptions. A set of numerical examples confirm the efficiency, robustness and reliability of the proposed numerical technique.
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... The basic filters will be obtained via an adaptive interpolatory procedure that aims to avoid blurred edges by not allowing the averaging of data ... An example of this strategy can be found in [1, 2 and 6], where a nonlinear ENO-type... more
... The basic filters will be obtained via an adaptive interpolatory procedure that aims to avoid blurred edges by not allowing the averaging of data ... An example of this strategy can be found in [1, 2 and 6], where a nonlinear ENO-type technique (Essentially Nonoscillatory, see [8]) is ...
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Abstract. We present a new method,for solving total variation (TV) minimization problems in image restoration. The main idea is to remove some of the singularity caused by the nondif- ferentiability of the quantity |∇u| in the definition... more
Abstract. We present a new method,for solving total variation (TV) minimization problems in image restoration. The main idea is to remove some of the singularity caused by the nondif- ferentiability of the quantity |∇u| in the definition of the TV-norm before we apply a linearization technique such as Newton’s method. This is accomplished by introducing an additional variable for the
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Research Interests: Applied Mathematics, Image, Image Restoration, Viscosity, Discontinuity, and 12 moreRegion, Edge, Convolution, PARTIAL DIFFERENTIAL EQUATION, Gaussian distribution, Euler Lagrange Equation, Numerical Analysis and Computational Mathematics, Waste minimisation, Boundary Condition, Linearity, Minimization, and Total Variation
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ABSTRACT Liu, Osher, and Chan introduced weighted essentially nonoscillatory (WENO) reconstructions in [X.-D. Liu, S. Osher, and T. Chan, J. Comput. Phys., 115 (1994), pp. 200–212] to improve the order of accuracy of essentially... more
ABSTRACT Liu, Osher, and Chan introduced weighted essentially nonoscillatory (WENO) reconstructions in [X.-D. Liu, S. Osher, and T. Chan, J. Comput. Phys., 115 (1994), pp. 200–212] to improve the order of accuracy of essentially nonoscillatory (ENO) reconstructions [A. Harten et al., J. Comput. Phys., 71 (1987), pp. 231–303]. In [G.-S. Jiang and C.-W. Shu, J. Comput. Phys., 126 (1996), pp. 202–228], the authors proposed smoothness indicators to obtain a WENO fifth order reconstruction from third order ENO reconstructions. With these smoothness indicators, Balsara and Shu [J. Comput. Phys., 160 (2000), pp. 405–452] and, later, [G. A. Gerolymos, D. Sénéchal, and I. Vallet, J. Comput. Phys., 228 (2009), pp. 8481–8524] obtained (2r − l)th order WENO reconstructions from rth order ENO reconstructions for 4 ≤ r ≤ 6, resp., 7 ≤ r ≤ 9. In [A. K. Henrick, T. D. Aslam, and J. M. Powers, J. Comput. Phys., 207 (2005), pp. 542–567], the authors noticed that these reconstructions do not attain the optimal order 2r − 1 at extrema and they proposed a fix for the problem. Other authors [R. Borges et al., J. Comput. Phys., 227 (2008), pp. 3191–3211; N. K. Yamaleev and M. H. Carpenter, J. Comput. Phys., 228 (2009), pp. 4248–4272] have addressed this problem with different weight designs for Jiang—Shu smoothness indicators. In this paper we exploit the special structure of Jiang-Shu smoothness indicators and analyze the role of a parameter appearing in the weight definition to avoid division by zero to obtain for any r > 2, by standard approximation properties of Lagrange interpolation, that the order of the WENO reconstruction is 2r − 1 at smooth regions, regardless of neighboring extrema, whilst this order is r, as ENO reconstructions have, when the function has a discontinuity in the stencil of 2r − 1 points but it is smooth in at least one of the substencils of r points. The optimal weights are also obtained in closed form.
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The ideas that lead from ENO to Weighted ENO (WENO) reconstructions (i.e. cell-average “interpolators”), devised and extensively used for the design of highly accurate shock capturing schemes for conservation laws, are applied in this... more
The ideas that lead from ENO to Weighted ENO (WENO) reconstructions (i.e. cell-average “interpolators”), devised and extensively used for the design of highly accurate shock capturing schemes for conservation laws, are applied in this paper to obtain weighted essentially non-oscillatory point-value nonlinear interpolators that can generically achieve an order of accuracy of 2r, when using stencils of 2r points at regions where the interpolated function is smooth. This interpolatory technique can be used in Harten’s multiresolution framework for image compression applications. More specifically, the nonlinear weights which the present interpolation is based upon are computed as proposed in Liu et al. (J. Comput. Phys. 115(1):200–212, 1994) and depend on smoothness indicators of the sub-stencils, defined in a way inspired by the smoothness indicators proposed in Jiang and Shu (J. Comput. Phys. 126(1):202–228, 1996), but through the corresponding Lagrange interpolators, instead of the cell-average interpolators. We setup a unified framework that eases the consecution of the following results for any r, when using stencils of 2r points, by only using properties of the Lagrange interpolators: (1) the order of the interpolation is 2r at smooth regions, regardless of neighboring extrema; this is true even around points where successive derivatives of the function vanish; (2) the order of the interpolation is r+1, like the ENO interpolants, when the function has a discontinuity in the stencil of 2r points but it is smooth in at least one of the sub-stencils of r+1 points; (3) the optimal weights are obtained in closed form. All these results are obtained by a thorough study that highlights the importance of setting the parameter ε that appears in the definition of the weights to avoid null denominators to ε=h 2. The image compression capability of this interpolation is compared to other standard image compression techniques to conclude that its strength can be found in applications where images have relatively large regions of smoothness.
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... conditions studied by Haas and Sturtevant [J. Fluid Mech. 181 (1987) 41] and successfully simulated by Quirk and Karni [J. Fluid Mech. 318 (1996) 129]. Author Keywords: Multiphase gas dynamics; RichtmyerMeshkov instability. ...
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A spatial-temporal transmission model of 2009 A/H1N1 pandemic influenza across Chile, a country that spans a large latitudinal range, is developed to characterize the spatial variation in peak timing of that pandemic as a function of... more
A spatial-temporal transmission model of 2009 A/H1N1 pandemic influenza across Chile, a country that spans a large latitudinal range, is developed to characterize the spatial variation in peak timing of that pandemic as a function of local transmission rates, spatial connectivity assumptions for Chilean regions, and the putative location of introduction of the novel virus into the country. Specifically, a metapopulation SEIR (susceptible-exposed-infected-removed) compartmental model that tracks the transmission dynamics of influenza in 15 Chilean regions is calibrated. The model incorporates population mobility among neighboring regions and indirect mobility to and from other regions via the metropolitan central region ('hub region'). The stability of the disease-free equilibrium of this model is analyzed and compared with the corresponding stability in each region, concluding that stability may occur even with some regions having basic reproduction numbers above 1. The transmission model is used along with epidemiological data to explore potential factors that could have driven the spatial-temporal progression of the pandemic. Simulations and sensitivity analyses indicate that this relatively simple model is sufficient to characterize the south-north gradient in peak timing observed during the pandemic, and suggest that south Chile observed the initial spread of the pandemic virus, which is in line with a retrospective epidemiological study. The 'hub region' in our model significantly enhanced population mixing in a short time scale.
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ABSTRACT Mathematical models of multi-phase flow are useful in some engineering applications like enhanced oil recovery, filtration of pollutants into subsurface, etc. In this work, we derive a mathematical model for the motion of... more
ABSTRACT Mathematical models of multi-phase flow are useful in some engineering applications like enhanced oil recovery, filtration of pollutants into subsurface, etc. In this work, we derive a mathematical model for the motion of one-dimensional three-phase flow in a porous medium under the condition of vertical equilibrium, which can be viewed as an extension of some two-phase flow models described in the literature. Our model involves a system of two partial differential equations in the form of viscous conservation laws, whose solutions may contain very sharp transitions. We show that a high-order/high resolution Weighted Essentially Non Oscillatory scheme is an appropriate tool to discretize the buoyancy flux and obtain a well resolved representation of the solution of the model. In addition, we show that the efficiency of the scheme may be improved by using Implicit–Explicit (IMEX) strategies, where the parabolic terms are handled by an implicit discretization.
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... to traditional methods and naturally a part of the research in this area is directed towards making them ... ear problem, as well as for the linear algebra problems which arise at each step. ... For simplicity, we will consider the... more
... to traditional methods and naturally a part of the research in this area is directed towards making them ... ear problem, as well as for the linear algebra problems which arise at each step. ... For simplicity, we will consider the unconstrained formulation of the Tikhonov regularization. ...
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ABSTRACT The motion of two-phase flow in a porous medium under the condition of vertical equilibrium can be described by a viscous conservation law that involves a non-convex flux function with two inflection points. In [5], a first order... more
ABSTRACT The motion of two-phase flow in a porous medium under the condition of vertical equilibrium can be described by a viscous conservation law that involves a non-convex flux function with two inflection points. In [5], a first order Godunov scheme was used to numerically approximate solutions of the model. In this paper we show that using instead the high resolutionWeighted Essentially Non Oscillatory (WENO) technology, and an IMEX strategy to handle the capillary term by an implicit discretization, leads to a noticeable increase in resolution power and efficiency. We carefully discuss the implementation of WENO schemes for the model equation, paying special attention to the choice of the definition of the numerical viscosity. We also present numerical simulations when the capillary number is negligible (i.e., the model is a homogeneous conservation law) and non-negligible (i.e. the model equation becomes a ’viscous’ conservation law). The numerical results are compared with those obtained with the method proposed in [5] in terms of accuracy, resolution power and global efficiency.
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Abstract. Adaptive mesh refinement algorithms were introduced in [2, 4] for the design of artificial viscosity schemes with nonuniform, adaptive, resolution on Cartesian meshes for hyperbolic conservation laws. These techniques have been... more
Abstract. Adaptive mesh refinement algorithms were introduced in [2, 4] for the design of artificial viscosity schemes with nonuniform, adaptive, resolution on Cartesian meshes for hyperbolic conservation laws. These techniques have been extended to finite volumes schemes in ...