We propose an enriched Galerkin-characteristics finite element method for numerical solution of c... more We propose an enriched Galerkin-characteristics finite element method for numerical solution of convection-dominated problems. The method uses the modified method of characteristics for the integration of the total derivative in time, combined with the finite element method for the spatial discretization on unstructured grids. The L 2-projection method is implemented for the evaluation of numerical solutions by tracking the departure points from the integration points in each element. In the present study, a family of quadrature rules are used to enrich the approximation of integrals in the L 2-projection method. The use of quadrature rules as an enrichment procedure allows for spatial discretizations on coarse fixed meshes and no need to introduce time-dependent enrichments. This procedure offers a very great advantage over the conventional Galerkin-characteristics finite element method since the same governing matrix representation can be used during the entire time steeping process. We also propose a multilevel adaptive procedure in the enriched Galerkin-characteristics finite element method by monitoring the gradient of the solution in the computational domain during its advection. In comparison with the traditional finite element analysis with h-, p-and hp-version refinements, the present approach is much simpler, more robust and efficient, and it yields more accurate solutions for a fixed number of degrees of freedom without refining the mesh. To examine the performance of the proposed method we solve several test examples for convection-diffusion problems. Comparison to the conventional Galerkin-characteristics finite element method is also carried out in the present work. The aim of such enriched method compared to the classical finite element method is to solve the time-dependent convection-dominated problems efficiently and with an appropriate level of accuracy.
A class of Bernstein-Bézier basis based high-order finite element methods is developed for the Ga... more A class of Bernstein-Bézier basis based high-order finite element methods is developed for the Galerkin-characteristics solution of convection-diffusion problems. The Galerkin-characteristics formulation is derived using a semi-Lagrangian discretization of the total derivative in the considered problems. The spatial discretization is performed using the finite element method on unstructured meshes. The Lagrangian interpretation in this approach greatly reduces the time truncation errors in the Eulerian methods. To achieve high-order accuracy in the Galerkin-characteristics solver, the semi-Lagrangian method requires high-order interpolating procedures. In the present work, this step is carried out using the Bernstein-Bézier basis functions to evaluate the solution at the departure points. Triangular Bernstein-Bézier patches are constructed in a simple and inherent manner over finite elements along the characteristics. An efficient preconditioned conjugate gradient solver is used for...
L'objectif de ce travail est l'étude d'une méthode de résolution numérique par éléments finis sem... more L'objectif de ce travail est l'étude d'une méthode de résolution numérique par éléments finis semilagrangiennes afin de résoudre les problémes évolutifs de convection-diffusion issus des milieux poreux. La méthode proposée permet d'utiliser une approximation par éléments finis d'ordre égal pour toutes les solutions du probléme. En outre, la condition standard de Courant-Friedrichs-Lewy est assouplie avec le traitement lagrangien des termes de convection, et les erreurs de troncature sont réduites dans la partie diffusion-réaction du probléme. Dans cette étude, une analyse de la convergence et de la stabilité de la méthode proposée est aussi présentée, ainsi que les estimations des erreurs dans la norme L 2 dérivées pour toutes les solutions numériques. Les tests numériques sont illustrées par quelques exemples afin de vérifier les estimations théoriques et de démontrer la grande précision et l'efficacité de la méthode proposée. ABSTRACT. We present a Galerkin-characteristic finite element method for the numerical solution of time-dependent convection-diffusion problems in porous media. The proposed method allows the use of equal-order finite element approximations for all solutions in the problem. In addition, the standard Courant-Friedrichs-Lewy condition is relaxed with the Lagrangian treatment of convection terms, and the time truncation errors are reduced in the diffusion-reaction part. Analysis of convergence and stability of the proposed method is also investigated in this study and error estimates in the L 2-norm are established for the numerical solutions. Numerical performance of the method is examined using two examples to verify the theoretical estimates and to demonstrate the high accuracy and efficiency of the proposed Galerkin-characteristic finite element method. MOTS-CLÉS. Problémes de convection-diffusion, Équation de Darcy, Milieux poreux, Méthode semi-lagrangienne, Éléments finis, Estimation des erreurs a priori.
International Journal of Mechanical and Mechatronics Engineering, Jun 5, 2020
Modeling dam-break flows over non-flat beds requires an accurate representation of the topography... more Modeling dam-break flows over non-flat beds requires an accurate representation of the topography which is the main source of uncertainty in the model. Therefore, developing robust and accurate techniques for reconstructing topography in this class of problems would reduce the uncertainty in the flow system. In many hydraulic applications, experimental techniques have been widely used to measure the bed topography. In practice, experimental work in hydraulics may be very demanding in both time and cost. Meanwhile, computational hydraulics have served as an alternative for laboratory and field experiments. Unlike the forward problem, the inverse problem is used to identify the bed parameters from the given experimental data. In this case, the shallow water equations used for modeling the hydraulics need to be rearranged in a way that the model parameters can be evaluated from measured data. However, this approach is not always possible and it suffers from stability restrictions. In the present work, we propose an adaptive optimal control technique to numerically identify the underlying bed topography from a given set of free-surface observation data. In this approach, a minimization function is defined to iteratively determine the model parameters. The proposed technique can be interpreted as a fractional-stage scheme. In the first stage, the forward problem is solved to determine the measurable parameters from known data. In the second stage, the adaptive control Ensemble Kalman Filter is implemented to combine the optimality of observation data in order to obtain the accurate estimation of the topography. The main features of this method are on one hand, the ability to solve for different complex geometries with no need for any rearrangements in the original model to rewrite it in an explicit form. On the other hand, its achievement of strong stability for simulations of flows in different regimes containing shocks or discontinuities over any geometry. Numerical results are presented for a dam-break flow problem over non-flat bed using different solvers for the shallow water equations. The robustness of the proposed method is investigated using different numbers of loops, sensitivity parameters, initial samples and location of observations. The obtained results demonstrate high reliability and accuracy of the proposed techniques.
A coupled two-layer finite volume/finite element<br> method was proposed for solving dam-br... more A coupled two-layer finite volume/finite element<br> method was proposed for solving dam-break flow problem<br> over deformable beds. The governing equations consist of the<br> well-balanced two-layer shallow water equations for the water flow<br> and a linear elastic model for the bed deformations. Deformations<br> in the topography can be caused by a brutal localized force or<br> simply by a class of sliding displacements on the bathymetry.<br> This deformation in the bed is a source of perturbations, on<br> the water surface generating water waves which propagate with<br> different amplitudes and frequencies. Coupling conditions at the<br> interface are also investigated in the current study and two mesh<br> procedure is proposed for the transfer of information through the<br> interface. In the present work a new procedure is implemented at<br> the soil-water interface using the finite element and two-l...
We present a new class of numerical techniques to<br> solve shallow water flows over dry ar... more We present a new class of numerical techniques to<br> solve shallow water flows over dry areas including run-up. Many<br> recent investigations on wave run-up in coastal areas are based on<br> the well-known shallow water equations. Numerical simulations have<br> also performed to understand the effects of several factors on tsunami<br> wave impact and run-up in the presence of coastal areas. In all these<br> simulations the shallow water equations are solved in entire domain<br> including dry areas and special treatments are used for numerical<br> solution of singularities at these dry regions. In the present study we<br> propose a new method to deal with these difficulties by reformulating<br> the shallow water equations into a new system to be solved only in the<br> wetted domain. The system is obtained by a change in the coordinates<br> leading to a set of equations in a moving domain for which the<br>...
A fast finite volume solver for multi-layered shallow<br> water flows with mass exchange an... more A fast finite volume solver for multi-layered shallow<br> water flows with mass exchange and an erodible bed is developed.<br> This enables the user to solve a number of complex sediment-based<br> problems including (but not limited to), dam-break over an erodible<br> bed, recirculation currents and bed evolution as well as levy and<br> dyke failure. This research develops methodologies crucial to the<br> under-standing of multi-sediment fluvial mechanics and waterway<br> design. In this model mass exchange between the layers is allowed<br> and, in contrast to previous models, sediment and fluid are able<br> to transfer between layers. In the current study we use a two-step<br> finite volume method to avoid the solution of the Riemann problem.<br> Entrainment and deposition rates are calculated for the first time in<br> a model of this nature. In the first step the governing equations are<br> rewritten in ...
This paper deals with the development of a stable and efficient unified finite element method for... more This paper deals with the development of a stable and efficient unified finite element method for the numerical solution of thermal Darcy flows with variable viscosity. The governing equations consist of coupling the Darcy equations for the pressure and velocity fields to a convection-diffusion equation for the heat transfer. The viscosity in the Darcy flows is assumed to be nonlinear depending on the temperature of the medium. The proposed method is based on combining a semi-Lagrangian scheme with a Galerkin finite element discretization of the governing equations along with an robust iterative solver for the associate linear systems. The main features of the enhanced finite element algorithm are that the same finite element space is used for all solutions to the problem including the pressure, velocity and temperature. In addition, the convection terms are accurately dealt with using the semi-Lagrangian scheme and the standard Courant-Friedrichs-Lewy condition is relaxed and the time truncation errors are reduced in the diffusion terms. Numerical results are presented for two examples to demonstrate the performance of the proposed finite element algorithm.
We develop a class of numerical methods for solving optimal control problems governed by nonlinea... more We develop a class of numerical methods for solving optimal control problems governed by nonlinear conservation laws in two space dimensions. The relaxation approximation is used to transform the nonlinear problem to a semi-linear diagonalizable system with source terms. The relaxing system is hyperbolic and it can be numerically solved without need to either Riemann solvers for space discretization or a non-linear system of algebraic equations solvers for time discretization. In the current study, the optimal control problem is formulated for the relaxation system and at the relaxed limit its solution converges to the relaxed equation of conservation laws. An upwind method is used for reconstruction of numerical fluxes and an implicit-explicit scheme is used for time stepping. Computational results are presented for a two-dimensional inviscid Burgers problem.
This is a PDF file of an article that has undergone enhancements after acceptance, such as the ad... more This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Journal of Computational and Applied Mathematics, 2020
We investigate the performance of a unified finite element method for the numerical solution of m... more We investigate the performance of a unified finite element method for the numerical solution of moving fronts in porous media under non-isothermal flow conditions. The governing equations consist of coupling the Darcy equation for the pressure to two convection-diffusion-reaction equations for the temperature and depth of conversion. The aim is to develop a non-oscillatory unified Galerkin-characteristic method for efficient simulation of moving fronts in porous media. The method is based on combining the modified method of characteristics with a Galerkin finite element discretization of the governing equations. The main feature of the proposed unified finite element method is that the same finite element space is used for all solutions to the problem including the pressure, velocity, temperature and concentration. Analysis of convergence and stability is also presented in this study and error estimates in the L 2-norm are established for the numerical solutions. In addition, due to the Lagrangian treatment of convection terms, the standard Courant-Friedrichs-Lewy condition is relaxed and the time truncation errors are reduced in the diffusion-reaction part. We verify the method for the benchmark problem of moving fronts around an array of cylinders. The numerical results obtained demonstrate the ability of the proposed method to capture the main flow features.
We propose an enriched Galerkin-characteristics finite element method for numerical solution of c... more We propose an enriched Galerkin-characteristics finite element method for numerical solution of convection-dominated problems. The method uses the modified method of characteristics for the integration of the total derivative in time, combined with the finite element method for the spatial discretization on unstructured grids. The L 2-projection method is implemented for the evaluation of numerical solutions by tracking the departure points from the integration points in each element. In the present study, a family of quadrature rules are used to enrich the approximation of integrals in the L 2-projection method. The use of quadrature rules as an enrichment procedure allows for spatial discretizations on coarse fixed meshes and no need to introduce time-dependent enrichments. This procedure offers a very great advantage over the conventional Galerkin-characteristics finite element method since the same governing matrix representation can be used during the entire time steeping process. We also propose a multilevel adaptive procedure in the enriched Galerkin-characteristics finite element method by monitoring the gradient of the solution in the computational domain during its advection. In comparison with the traditional finite element analysis with h-, p-and hp-version refinements, the present approach is much simpler, more robust and efficient, and it yields more accurate solutions for a fixed number of degrees of freedom without refining the mesh. To examine the performance of the proposed method we solve several test examples for convection-diffusion problems. Comparison to the conventional Galerkin-characteristics finite element method is also carried out in the present work. The aim of such enriched method compared to the classical finite element method is to solve the time-dependent convection-dominated problems efficiently and with an appropriate level of accuracy.
A class of Bernstein-Bézier basis based high-order finite element methods is developed for the Ga... more A class of Bernstein-Bézier basis based high-order finite element methods is developed for the Galerkin-characteristics solution of convection-diffusion problems. The Galerkin-characteristics formulation is derived using a semi-Lagrangian discretization of the total derivative in the considered problems. The spatial discretization is performed using the finite element method on unstructured meshes. The Lagrangian interpretation in this approach greatly reduces the time truncation errors in the Eulerian methods. To achieve high-order accuracy in the Galerkin-characteristics solver, the semi-Lagrangian method requires high-order interpolating procedures. In the present work, this step is carried out using the Bernstein-Bézier basis functions to evaluate the solution at the departure points. Triangular Bernstein-Bézier patches are constructed in a simple and inherent manner over finite elements along the characteristics. An efficient preconditioned conjugate gradient solver is used for...
L'objectif de ce travail est l'étude d'une méthode de résolution numérique par éléments finis sem... more L'objectif de ce travail est l'étude d'une méthode de résolution numérique par éléments finis semilagrangiennes afin de résoudre les problémes évolutifs de convection-diffusion issus des milieux poreux. La méthode proposée permet d'utiliser une approximation par éléments finis d'ordre égal pour toutes les solutions du probléme. En outre, la condition standard de Courant-Friedrichs-Lewy est assouplie avec le traitement lagrangien des termes de convection, et les erreurs de troncature sont réduites dans la partie diffusion-réaction du probléme. Dans cette étude, une analyse de la convergence et de la stabilité de la méthode proposée est aussi présentée, ainsi que les estimations des erreurs dans la norme L 2 dérivées pour toutes les solutions numériques. Les tests numériques sont illustrées par quelques exemples afin de vérifier les estimations théoriques et de démontrer la grande précision et l'efficacité de la méthode proposée. ABSTRACT. We present a Galerkin-characteristic finite element method for the numerical solution of time-dependent convection-diffusion problems in porous media. The proposed method allows the use of equal-order finite element approximations for all solutions in the problem. In addition, the standard Courant-Friedrichs-Lewy condition is relaxed with the Lagrangian treatment of convection terms, and the time truncation errors are reduced in the diffusion-reaction part. Analysis of convergence and stability of the proposed method is also investigated in this study and error estimates in the L 2-norm are established for the numerical solutions. Numerical performance of the method is examined using two examples to verify the theoretical estimates and to demonstrate the high accuracy and efficiency of the proposed Galerkin-characteristic finite element method. MOTS-CLÉS. Problémes de convection-diffusion, Équation de Darcy, Milieux poreux, Méthode semi-lagrangienne, Éléments finis, Estimation des erreurs a priori.
International Journal of Mechanical and Mechatronics Engineering, Jun 5, 2020
Modeling dam-break flows over non-flat beds requires an accurate representation of the topography... more Modeling dam-break flows over non-flat beds requires an accurate representation of the topography which is the main source of uncertainty in the model. Therefore, developing robust and accurate techniques for reconstructing topography in this class of problems would reduce the uncertainty in the flow system. In many hydraulic applications, experimental techniques have been widely used to measure the bed topography. In practice, experimental work in hydraulics may be very demanding in both time and cost. Meanwhile, computational hydraulics have served as an alternative for laboratory and field experiments. Unlike the forward problem, the inverse problem is used to identify the bed parameters from the given experimental data. In this case, the shallow water equations used for modeling the hydraulics need to be rearranged in a way that the model parameters can be evaluated from measured data. However, this approach is not always possible and it suffers from stability restrictions. In the present work, we propose an adaptive optimal control technique to numerically identify the underlying bed topography from a given set of free-surface observation data. In this approach, a minimization function is defined to iteratively determine the model parameters. The proposed technique can be interpreted as a fractional-stage scheme. In the first stage, the forward problem is solved to determine the measurable parameters from known data. In the second stage, the adaptive control Ensemble Kalman Filter is implemented to combine the optimality of observation data in order to obtain the accurate estimation of the topography. The main features of this method are on one hand, the ability to solve for different complex geometries with no need for any rearrangements in the original model to rewrite it in an explicit form. On the other hand, its achievement of strong stability for simulations of flows in different regimes containing shocks or discontinuities over any geometry. Numerical results are presented for a dam-break flow problem over non-flat bed using different solvers for the shallow water equations. The robustness of the proposed method is investigated using different numbers of loops, sensitivity parameters, initial samples and location of observations. The obtained results demonstrate high reliability and accuracy of the proposed techniques.
A coupled two-layer finite volume/finite element<br> method was proposed for solving dam-br... more A coupled two-layer finite volume/finite element<br> method was proposed for solving dam-break flow problem<br> over deformable beds. The governing equations consist of the<br> well-balanced two-layer shallow water equations for the water flow<br> and a linear elastic model for the bed deformations. Deformations<br> in the topography can be caused by a brutal localized force or<br> simply by a class of sliding displacements on the bathymetry.<br> This deformation in the bed is a source of perturbations, on<br> the water surface generating water waves which propagate with<br> different amplitudes and frequencies. Coupling conditions at the<br> interface are also investigated in the current study and two mesh<br> procedure is proposed for the transfer of information through the<br> interface. In the present work a new procedure is implemented at<br> the soil-water interface using the finite element and two-l...
We present a new class of numerical techniques to<br> solve shallow water flows over dry ar... more We present a new class of numerical techniques to<br> solve shallow water flows over dry areas including run-up. Many<br> recent investigations on wave run-up in coastal areas are based on<br> the well-known shallow water equations. Numerical simulations have<br> also performed to understand the effects of several factors on tsunami<br> wave impact and run-up in the presence of coastal areas. In all these<br> simulations the shallow water equations are solved in entire domain<br> including dry areas and special treatments are used for numerical<br> solution of singularities at these dry regions. In the present study we<br> propose a new method to deal with these difficulties by reformulating<br> the shallow water equations into a new system to be solved only in the<br> wetted domain. The system is obtained by a change in the coordinates<br> leading to a set of equations in a moving domain for which the<br>...
A fast finite volume solver for multi-layered shallow<br> water flows with mass exchange an... more A fast finite volume solver for multi-layered shallow<br> water flows with mass exchange and an erodible bed is developed.<br> This enables the user to solve a number of complex sediment-based<br> problems including (but not limited to), dam-break over an erodible<br> bed, recirculation currents and bed evolution as well as levy and<br> dyke failure. This research develops methodologies crucial to the<br> under-standing of multi-sediment fluvial mechanics and waterway<br> design. In this model mass exchange between the layers is allowed<br> and, in contrast to previous models, sediment and fluid are able<br> to transfer between layers. In the current study we use a two-step<br> finite volume method to avoid the solution of the Riemann problem.<br> Entrainment and deposition rates are calculated for the first time in<br> a model of this nature. In the first step the governing equations are<br> rewritten in ...
This paper deals with the development of a stable and efficient unified finite element method for... more This paper deals with the development of a stable and efficient unified finite element method for the numerical solution of thermal Darcy flows with variable viscosity. The governing equations consist of coupling the Darcy equations for the pressure and velocity fields to a convection-diffusion equation for the heat transfer. The viscosity in the Darcy flows is assumed to be nonlinear depending on the temperature of the medium. The proposed method is based on combining a semi-Lagrangian scheme with a Galerkin finite element discretization of the governing equations along with an robust iterative solver for the associate linear systems. The main features of the enhanced finite element algorithm are that the same finite element space is used for all solutions to the problem including the pressure, velocity and temperature. In addition, the convection terms are accurately dealt with using the semi-Lagrangian scheme and the standard Courant-Friedrichs-Lewy condition is relaxed and the time truncation errors are reduced in the diffusion terms. Numerical results are presented for two examples to demonstrate the performance of the proposed finite element algorithm.
We develop a class of numerical methods for solving optimal control problems governed by nonlinea... more We develop a class of numerical methods for solving optimal control problems governed by nonlinear conservation laws in two space dimensions. The relaxation approximation is used to transform the nonlinear problem to a semi-linear diagonalizable system with source terms. The relaxing system is hyperbolic and it can be numerically solved without need to either Riemann solvers for space discretization or a non-linear system of algebraic equations solvers for time discretization. In the current study, the optimal control problem is formulated for the relaxation system and at the relaxed limit its solution converges to the relaxed equation of conservation laws. An upwind method is used for reconstruction of numerical fluxes and an implicit-explicit scheme is used for time stepping. Computational results are presented for a two-dimensional inviscid Burgers problem.
This is a PDF file of an article that has undergone enhancements after acceptance, such as the ad... more This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Journal of Computational and Applied Mathematics, 2020
We investigate the performance of a unified finite element method for the numerical solution of m... more We investigate the performance of a unified finite element method for the numerical solution of moving fronts in porous media under non-isothermal flow conditions. The governing equations consist of coupling the Darcy equation for the pressure to two convection-diffusion-reaction equations for the temperature and depth of conversion. The aim is to develop a non-oscillatory unified Galerkin-characteristic method for efficient simulation of moving fronts in porous media. The method is based on combining the modified method of characteristics with a Galerkin finite element discretization of the governing equations. The main feature of the proposed unified finite element method is that the same finite element space is used for all solutions to the problem including the pressure, velocity, temperature and concentration. Analysis of convergence and stability is also presented in this study and error estimates in the L 2-norm are established for the numerical solutions. In addition, due to the Lagrangian treatment of convection terms, the standard Courant-Friedrichs-Lewy condition is relaxed and the time truncation errors are reduced in the diffusion-reaction part. We verify the method for the benchmark problem of moving fronts around an array of cylinders. The numerical results obtained demonstrate the ability of the proposed method to capture the main flow features.
We are honoured to bring you this book dedicated to recent advances on applied mathematics and st... more We are honoured to bring you this book dedicated to recent advances on applied mathematics and statistics. These articles comprise a diverse collection of theoretical results as well as applications, primarily centred on mathematical modelling, applied mathematics and statistics. The papers in this book were presented in the 1st Hispano-Moroccan Days on Applied Mathematics and Statistics held at the Faculte des Sciences, Tetouan, Morocco in December 2008. This conference is co-organized by the Universidad Rey Juan Carlos, Madrid, Spain, and Universite Abdelmalek Essaadi, Faculte des Sciences de Tetouan, Morocco. The reader will find in this book analysis of partial differential equations, networks, linear algebra, and statistical methods. There are articles on fluid dynamics, both modelling and computational, finite element and finite volume methods and stochastic partial differential equations, as well as various applications in probability and control theory. We have enjoyed the review process for these articles, as they are inspirational for new directions within applied mathematics and statistics.
In this work, we study the existence of compact almost automorphic solutions for a class of integ... more In this work, we study the existence of compact almost automorphic solutions for a class of integral equations with a time-dependent delay and a history-dependent delay. An application to a blow ‡ies model and a transmission lines model is carried out to support the theoretical …nding.
The intrinsic concept of meshless methods may be found in many approaches in interpolation and nu... more The intrinsic concept of meshless methods may be found in many approaches in interpolation and numerical methods for partial differential equations. Given this common concept, the aim of the Euro-Mediterranean workshop is to provide an opportunity for researchers and practitioners to discuss recent research results that may support a wide applicability in meshless related approaches. To build a foundation for these discussions, a number of experts has been invited to talk about their research. The covered topics will range from theoretical analysis of meshless based methods to applications and aspects of implementation. Anybody interested to participate is cordially invited to join the discussions and to give a presentation on his/her own research.
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