Cockburn, B. (Bernardo)
Overview
Works:  74 works in 163 publications in 3 languages and 1,603 library holdings 

Genres:  Conference papers and proceedings 
Roles:  Author, Contributor, Editor, htt, Collector, Other 
Classifications:  QA3, 620.00151535 
Publication Timeline
.
Most widely held works by
B Cockburn
Advanced numerical approximation of nonlinear hyperbolic equations : lectures given at the 2nd session of the Centro Internazionale
Matematico Estivo (C.I.M.E.) held in Cetraro, Italy, June 2328, 1997 by
B Cockburn(
Book
)
30 editions published between 1998 and 2006 in English and held by 581 WorldCat member libraries worldwide
This volume contains the texts of the four series of lectures presented by B.Cockburn, C.Johnson, C.W. Shu and E.Tadmor at a C.I.M.E. Summer School. It is aimed at providing a comprehensive and uptodate presentation of numerical methods which are nowadays used to solve nonlinear partial differential equations of hyperbolic type, developing shock discontinuities. The most effective methodologies in the framework of finite elements, finite differences, finite volumes spectral methods and kinetic methods, are addressed, in particular highorder shock capturing techniques, discontinuous Galerkin methods, adaptive techniques based upon aposteriori error analysis
30 editions published between 1998 and 2006 in English and held by 581 WorldCat member libraries worldwide
This volume contains the texts of the four series of lectures presented by B.Cockburn, C.Johnson, C.W. Shu and E.Tadmor at a C.I.M.E. Summer School. It is aimed at providing a comprehensive and uptodate presentation of numerical methods which are nowadays used to solve nonlinear partial differential equations of hyperbolic type, developing shock discontinuities. The most effective methodologies in the framework of finite elements, finite differences, finite volumes spectral methods and kinetic methods, are addressed, in particular highorder shock capturing techniques, discontinuous Galerkin methods, adaptive techniques based upon aposteriori error analysis
Discontinuous Galerkin methods : theory, computation, and applications by
B Cockburn(
Book
)
13 editions published between 2000 and 2009 in English and German and held by 283 WorldCat member libraries worldwide
This volume contains current progress of a new class of finite element method, the Discontinuous Galerkin Method (DGM), which has been under rapid developments recently and has found its use very quickly in such diverse applications as aeroacoustics, semiconductor device simulation, turbomachinery, turbulent flows, materials processing, Magnetohydrodynamics, plasma simulations and image processing. While there has been a lot of interest from mathematicians, physicists and engineers in DGM, only scattered information is available and there has been no prior effect in organizing and publishing the existing volume of knowledge on this subject. The current volume organizes this knowledge and it covers both theoretical as well as practical issues of the Discontinuous Galerkin method
13 editions published between 2000 and 2009 in English and German and held by 283 WorldCat member libraries worldwide
This volume contains current progress of a new class of finite element method, the Discontinuous Galerkin Method (DGM), which has been under rapid developments recently and has found its use very quickly in such diverse applications as aeroacoustics, semiconductor device simulation, turbomachinery, turbulent flows, materials processing, Magnetohydrodynamics, plasma simulations and image processing. While there has been a lot of interest from mathematicians, physicists and engineers in DGM, only scattered information is available and there has been no prior effect in organizing and publishing the existing volume of knowledge on this subject. The current volume organizes this knowledge and it covers both theoretical as well as practical issues of the Discontinuous Galerkin method
The P¹RKDG method for twodimensional Euler equations of gas dynamics by
B Cockburn(
Book
)
2 editions published in 1991 in English and held by 134 WorldCat member libraries worldwide
Earlier work is continued on a class on nonlinearly stable Runge Kutta local projection discontinuous Galerkin (RKDG) finite element methods for conservation laws. Two dimensional Euler equations for gas dynamics are solved using p1 elements. We discuss the generalization of the local projection, which for scalar nonlinear conservation laws was designed to satisfy a local maximum principle, to systems of conservation laws such as the Euler equations of gas dynamics using local characteristic decompositions. Numerical examples include the standard regular shock reflection problem, the forward facing step problem and the double Mach reflection problem. These preliminary numerical examples are chosen to show the capacity of our approach to obtain nonlinearly stable results comparable with the modern nonoscillatory finite difference methods. Generalizations to pk elements with k equal to or greater than the use of adaptive triangulations to minimize local errors constitute ongoing research
2 editions published in 1991 in English and held by 134 WorldCat member libraries worldwide
Earlier work is continued on a class on nonlinearly stable Runge Kutta local projection discontinuous Galerkin (RKDG) finite element methods for conservation laws. Two dimensional Euler equations for gas dynamics are solved using p1 elements. We discuss the generalization of the local projection, which for scalar nonlinear conservation laws was designed to satisfy a local maximum principle, to systems of conservation laws such as the Euler equations of gas dynamics using local characteristic decompositions. Numerical examples include the standard regular shock reflection problem, the forward facing step problem and the double Mach reflection problem. These preliminary numerical examples are chosen to show the capacity of our approach to obtain nonlinearly stable results comparable with the modern nonoscillatory finite difference methods. Generalizations to pk elements with k equal to or greater than the use of adaptive triangulations to minimize local errors constitute ongoing research
RungeKutta discontinuous Galerkin methods for convectiondominated problems by
B Cockburn(
Book
)
4 editions published in 2000 in English and held by 105 WorldCat member libraries worldwide
4 editions published in 2000 in English and held by 105 WorldCat member libraries worldwide
Enhanced accuracy by postprocessing for finite element methods for hyperbolic equations by
B Cockburn(
Book
)
4 editions published in 2000 in English and held by 103 WorldCat member libraries worldwide
4 editions published in 2000 in English and held by 103 WorldCat member libraries worldwide
Nonlinearly stable compact schemes for shock calculations by
B Cockburn(
Book
)
5 editions published in 1992 in English and held by 94 WorldCat member libraries worldwide
In this paper we discuss the applications of high order compact finite difference methods for shock calculations. The main idea is the definition of a local mean which serves as a reference for introducing a local nonlinear limiting to control spurious numerical oscillations while keeping the formal accuracy of the scheme. For scalar conservation laws, the resulting schemes can be proven total variation stable in one space dimension and maximum norm stable in multiple space dimensions. Numerical examples are shown to verify accuracy and stability of such schemes for problems containing shocks. The idea in this paper can also be applied to other implicit schemes such as the continuous Galerkin finite element methods
5 editions published in 1992 in English and held by 94 WorldCat member libraries worldwide
In this paper we discuss the applications of high order compact finite difference methods for shock calculations. The main idea is the definition of a local mean which serves as a reference for introducing a local nonlinear limiting to control spurious numerical oscillations while keeping the formal accuracy of the scheme. For scalar conservation laws, the resulting schemes can be proven total variation stable in one space dimension and maximum norm stable in multiple space dimensions. Numerical examples are shown to verify accuracy and stability of such schemes for problems containing shocks. The idea in this paper can also be applied to other implicit schemes such as the continuous Galerkin finite element methods
The Local Discontinuous Galerkin method for timedependent convectiondiffusion systems by
B Cockburn(
Book
)
6 editions published in 1997 in English and held by 93 WorldCat member libraries worldwide
In this paper, we study the Local Discontinuous Galerkin methods for nonlinear, timedependent convectiondiffusion systems. These methods are an extension of the RungeKutta Discontinuous Galerkin methods for purely hyperbolic systems to convectiondiffusion systems and share with those methods their high parallelizability, their highorder formal accuracy, and their easy handling of complicated geometries, for convection dominated problems. It is proven that for scalar equations, the Local Discontinuous Galerkin methods are L2stable in the nonlinear case. Moreover, in the linear case, it is shown that if polynomials of degree k are used, the methods are kth order accurate for general triangulations; although this order of convergence is suboptimal, it is sharp for the LDG methods. Preliminary numerical examples displaying the performance of the method are shown
6 editions published in 1997 in English and held by 93 WorldCat member libraries worldwide
In this paper, we study the Local Discontinuous Galerkin methods for nonlinear, timedependent convectiondiffusion systems. These methods are an extension of the RungeKutta Discontinuous Galerkin methods for purely hyperbolic systems to convectiondiffusion systems and share with those methods their high parallelizability, their highorder formal accuracy, and their easy handling of complicated geometries, for convection dominated problems. It is proven that for scalar equations, the Local Discontinuous Galerkin methods are L2stable in the nonlinear case. Moreover, in the linear case, it is shown that if polynomials of degree k are used, the methods are kth order accurate for general triangulations; although this order of convergence is suboptimal, it is sharp for the LDG methods. Preliminary numerical examples displaying the performance of the method are shown
The RungeKutta discontinuous Galerkin method for conservation laws V : multidimensional systems by
B Cockburn(
Book
)
5 editions published in 1997 in English and held by 90 WorldCat member libraries worldwide
This is the fifth paper in a series in which we construct and study the socalled RangeKutta Discontinuous Galerkin method for numerically solving hyperbolic conservation laws. In this paper, we extend the method to multidimensional nonlinear systems of conservation laws. The algorithms are described and discussed, including algorithm formulation and practical implementation issues such as the numerical fluxes, quadrature rules, degrees of freedom, and the slope limiters, both in the triangular and the rectangular element cases. Numerical experiments for two dimensional Euler equations of compressible gas dynamics are presented that show the effect of the (formal) order of accuracy and the use of triangles or rectangles, on the quality of the approximation
5 editions published in 1997 in English and held by 90 WorldCat member libraries worldwide
This is the fifth paper in a series in which we construct and study the socalled RangeKutta Discontinuous Galerkin method for numerically solving hyperbolic conservation laws. In this paper, we extend the method to multidimensional nonlinear systems of conservation laws. The algorithms are described and discussed, including algorithm formulation and practical implementation issues such as the numerical fluxes, quadrature rules, degrees of freedom, and the slope limiters, both in the triangular and the rectangular element cases. Numerical experiments for two dimensional Euler equations of compressible gas dynamics are presented that show the effect of the (formal) order of accuracy and the use of triangles or rectangles, on the quality of the approximation
Étude mathématique et numérique des équations de Maxwell dans des milieux polarisables by
B Cockburn(
)
5 editions published between 1983 and 2019 in French and held by 10 WorldCat member libraries worldwide
Formulation du problème par l'équation intégrale de Volterra. Approximation du noyau de convolation par un opérateur différentiel. Analyse numérique; stabilité et convergence du schéma numérique
5 editions published between 1983 and 2019 in French and held by 10 WorldCat member libraries worldwide
Formulation du problème par l'équation intégrale de Volterra. Approximation du noyau de convolation par un opérateur différentiel. Analyse numérique; stabilité et convergence du schéma numérique
Discontinuous Galerkin Methods : Theory, Computation and Applications(
)
in English and held by 7 WorldCat member libraries worldwide
in English and held by 7 WorldCat member libraries worldwide
The local discontinuous Galerkin method for linear incompressible fluid flow a review by
B Cockburn(
Book
)
2 editions published in 2003 in English and held by 7 WorldCat member libraries worldwide
2 editions published in 2003 in English and held by 7 WorldCat member libraries worldwide
A locally conservative LDG method for the incompressible Navier Stokes equations by
B Cockburn(
Book
)
2 editions published in 2004 in English and held by 7 WorldCat member libraries worldwide
2 editions published in 2004 in English and held by 7 WorldCat member libraries worldwide
Consistance et stabilite des schemas LRG pour les lois de conservation scalaires by
Guy Chavent(
Book
)
3 editions published in 1987 in French and held by 6 WorldCat member libraries worldwide
3 editions published in 1987 in French and held by 6 WorldCat member libraries worldwide
Le schéma Gk/2 pour les lois de conservation scalaires by
B Cockburn(
Book
)
4 editions published in 1985 in French and held by 4 WorldCat member libraries worldwide
4 editions published in 1985 in French and held by 4 WorldCat member libraries worldwide
A note on discontinuous Galerkin divergencefree solutions of the NavierStokes equations by
B Cockburn(
)
1 edition published in 2006 in English and held by 3 WorldCat member libraries worldwide
1 edition published in 2006 in English and held by 3 WorldCat member libraries worldwide
On convergence to entropy solutions of a single conservation law by
B Cockburn(
Book
)
3 editions published in 1995 in English and held by 3 WorldCat member libraries worldwide
3 editions published in 1995 in English and held by 3 WorldCat member libraries worldwide
StormerNumerov HDG Methods for Acoustic Waves by
B Cockburn(
)
1 edition published in 2017 in English and held by 2 WorldCat member libraries worldwide
1 edition published in 2017 in English and held by 2 WorldCat member libraries worldwide
The PlRKDG method for twodimensional Euler equations of gas dynamics by
B Cockburn(
Book
)
1 edition published in 1991 in English and held by 2 WorldCat member libraries worldwide
1 edition published in 1991 in English and held by 2 WorldCat member libraries worldwide
Nonlinearly stable compact schemes for shock calculations by
B Cockburn(
Book
)
2 editions published in 1992 in English and held by 2 WorldCat member libraries worldwide
2 editions published in 1992 in English and held by 2 WorldCat member libraries worldwide
Une Méthode d'éléments finis pour la simulation dans un réservoir de déplacements bidimensionnels d'huile par de l'eau by
Institut national de recherche en informatique et en automatique (France)(
Book
)
2 editions published in 1985 in French and held by 2 WorldCat member libraries worldwide
2 editions published in 1985 in French and held by 2 WorldCat member libraries worldwide
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Audience Level
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Related Identities
 Shu, ChiWang Collector Editor
 Quarteroni, Alfio Other Author Editor
 Centro internazionale matematico estivo
 Institute for Computer Applications in Science and Engineering
 Langley Research Center
 Karniadakis, George Collector Editor
 Johnson, C.
 Centro internazionale matematico estivo Session 1997 : Cetraro, Italie)
 Tadmor, Eitan
 University of Minnesota Institute for Mathematics and Its Applications
Associated Subjects
Algorithms Computational intelligence Computer science Computer scienceMathematics Conservation laws (Mathematics) Convergence Differential equations Differential equations, Hyperbolic Differential equations, HyperbolicNumerical solutions Differential equations, NonlinearNumerical solutions Differential equations, Partial Engineering Exponential functions Finite element method Galerkin methods Gas dynamics Mathematical physics Mathematics Maximum principles (Mathematics) Nonlinear systems Numerical analysis RungeKutta formulas Thermodynamics