Polynomial Chaos Methods for Hyperbolic Partial Differential Equations

Polynomial Chaos Methods for Hyperbolic Partial Differential Equations

Author: Mass Per Pettersson

Publisher: Springer

Published: 2015-03-10

Total Pages: 217

ISBN-13: 3319107143

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This monograph presents computational techniques and numerical analysis to study conservation laws under uncertainty using the stochastic Galerkin formulation. With the continual growth of computer power, these methods are becoming increasingly popular as an alternative to more classical sampling-based techniques. The text takes advantage of stochastic Galerkin projections applied to the original conservation laws to produce a large system of modified partial differential equations, the solutions to which directly provide a full statistical characterization of the effect of uncertainties. Polynomial Chaos Methods of Hyperbolic Partial Differential Equations focuses on the analysis of stochastic Galerkin systems obtained for linear and non-linear convection-diffusion equations and for a systems of conservation laws; a detailed well-posedness and accuracy analysis is presented to enable the design of robust and stable numerical methods. The exposition is restricted to one spatial dimension and one uncertain parameter as its extension is conceptually straightforward. The numerical methods designed guarantee that the solutions to the uncertainty quantification systems will converge as the mesh size goes to zero. Examples from computational fluid dynamics are presented together with numerical methods suitable for the problem at hand: stable high-order finite-difference methods based on summation-by-parts operators for smooth problems, and robust shock-capturing methods for highly nonlinear problems. Academics and graduate students interested in computational fluid dynamics and uncertainty quantification will find this book of interest. Readers are expected to be familiar with the fundamentals of numerical analysis. Some background in stochastic methods is useful but notnecessary.


Book Synopsis Polynomial Chaos Methods for Hyperbolic Partial Differential Equations by : Mass Per Pettersson

Download or read book Polynomial Chaos Methods for Hyperbolic Partial Differential Equations written by Mass Per Pettersson and published by Springer. This book was released on 2015-03-10 with total page 217 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph presents computational techniques and numerical analysis to study conservation laws under uncertainty using the stochastic Galerkin formulation. With the continual growth of computer power, these methods are becoming increasingly popular as an alternative to more classical sampling-based techniques. The text takes advantage of stochastic Galerkin projections applied to the original conservation laws to produce a large system of modified partial differential equations, the solutions to which directly provide a full statistical characterization of the effect of uncertainties. Polynomial Chaos Methods of Hyperbolic Partial Differential Equations focuses on the analysis of stochastic Galerkin systems obtained for linear and non-linear convection-diffusion equations and for a systems of conservation laws; a detailed well-posedness and accuracy analysis is presented to enable the design of robust and stable numerical methods. The exposition is restricted to one spatial dimension and one uncertain parameter as its extension is conceptually straightforward. The numerical methods designed guarantee that the solutions to the uncertainty quantification systems will converge as the mesh size goes to zero. Examples from computational fluid dynamics are presented together with numerical methods suitable for the problem at hand: stable high-order finite-difference methods based on summation-by-parts operators for smooth problems, and robust shock-capturing methods for highly nonlinear problems. Academics and graduate students interested in computational fluid dynamics and uncertainty quantification will find this book of interest. Readers are expected to be familiar with the fundamentals of numerical analysis. Some background in stochastic methods is useful but notnecessary.

Regularity and Generalized Polynomial Chaos Approximation of Parametric and Random 2nd Order Hyperbolic Partial Differential Equations

Regularity and Generalized Polynomial Chaos Approximation of Parametric and Random 2nd Order Hyperbolic Partial Differential Equations

Author: Viet Ha Hoang

Publisher:

Published: 2011

Total Pages:

ISBN-13:

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Book Synopsis Regularity and Generalized Polynomial Chaos Approximation of Parametric and Random 2nd Order Hyperbolic Partial Differential Equations by : Viet Ha Hoang

Download or read book Regularity and Generalized Polynomial Chaos Approximation of Parametric and Random 2nd Order Hyperbolic Partial Differential Equations written by Viet Ha Hoang and published by . This book was released on 2011 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Uncertainty Quantification for Hyperbolic and Kinetic Equations

Uncertainty Quantification for Hyperbolic and Kinetic Equations

Author: Shi Jin

Publisher: Springer

Published: 2018-03-20

Total Pages: 277

ISBN-13: 3319671103

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This book explores recent advances in uncertainty quantification for hyperbolic, kinetic, and related problems. The contributions address a range of different aspects, including: polynomial chaos expansions, perturbation methods, multi-level Monte Carlo methods, importance sampling, and moment methods. The interest in these topics is rapidly growing, as their applications have now expanded to many areas in engineering, physics, biology and the social sciences. Accordingly, the book provides the scientific community with a topical overview of the latest research efforts.


Book Synopsis Uncertainty Quantification for Hyperbolic and Kinetic Equations by : Shi Jin

Download or read book Uncertainty Quantification for Hyperbolic and Kinetic Equations written by Shi Jin and published by Springer. This book was released on 2018-03-20 with total page 277 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explores recent advances in uncertainty quantification for hyperbolic, kinetic, and related problems. The contributions address a range of different aspects, including: polynomial chaos expansions, perturbation methods, multi-level Monte Carlo methods, importance sampling, and moment methods. The interest in these topics is rapidly growing, as their applications have now expanded to many areas in engineering, physics, biology and the social sciences. Accordingly, the book provides the scientific community with a topical overview of the latest research efforts.

High Order Methods for Hyperbolic PDEs with Singular Source Term℗

High Order Methods for Hyperbolic PDEs with Singular Source Term℗

Author: Debananda Chakraborty

Publisher:

Published: 2012

Total Pages: 196

ISBN-13:

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In this research we consider hyperbolic partial differential equations with singular source term. First we consider the Zerilli equation, which models the phenomenon, when a star or other celestial object colloids with a black hole. In this model, there is no angular momentum. We develop the spectral-finite difference hybrid method which solves the equation very efficiently and accurately yields the quasi-normal modes and the power-law decay profile. This method is very fast compared to other methods. We also consider the sine-Gordon and nonlinear Schroedinger equations with a point-like singular source term. The soliton interaction with such a singular potential yields a critical solution behavior. That is, for the given value of the potential strength or the soliton amplitude, there exists a critical velocity of the initial soliton solution, around which the solution is either trapped by or transmitted through the potential.^In this research, we propose an efficient method for finding such a critical velocity by using the generalized polynomial chaos (gPC) method. For the proposed method, we assume that the soliton velocity is a random variable and expand the solution in the random space using the orthogonal polynomials. We consider the Legendre and Hermite chaos with both the Galerkin and collocation formulations. The proposed method finds the critical velocity accurately with spectral convergence. Thus the computational complexity is much reduced. The very core of the proposed method lies in using the mean solution instead of reconstructing the solution. The mean solution converges exponentially while the gPC reconstruction may fail to converge to the right solution due to the Gibbs phenomenon in the random space. Numerical results confirm the accuracy and spectral convergence of the method.^For the last problem a hybrid method based on the spectral method and weighted essentially non-oscillatory (WENO) finite difference method is proposed to solve the unsteady transonic equations.


Book Synopsis High Order Methods for Hyperbolic PDEs with Singular Source Term℗ by : Debananda Chakraborty

Download or read book High Order Methods for Hyperbolic PDEs with Singular Source Term℗ written by Debananda Chakraborty and published by . This book was released on 2012 with total page 196 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this research we consider hyperbolic partial differential equations with singular source term. First we consider the Zerilli equation, which models the phenomenon, when a star or other celestial object colloids with a black hole. In this model, there is no angular momentum. We develop the spectral-finite difference hybrid method which solves the equation very efficiently and accurately yields the quasi-normal modes and the power-law decay profile. This method is very fast compared to other methods. We also consider the sine-Gordon and nonlinear Schroedinger equations with a point-like singular source term. The soliton interaction with such a singular potential yields a critical solution behavior. That is, for the given value of the potential strength or the soliton amplitude, there exists a critical velocity of the initial soliton solution, around which the solution is either trapped by or transmitted through the potential.^In this research, we propose an efficient method for finding such a critical velocity by using the generalized polynomial chaos (gPC) method. For the proposed method, we assume that the soliton velocity is a random variable and expand the solution in the random space using the orthogonal polynomials. We consider the Legendre and Hermite chaos with both the Galerkin and collocation formulations. The proposed method finds the critical velocity accurately with spectral convergence. Thus the computational complexity is much reduced. The very core of the proposed method lies in using the mean solution instead of reconstructing the solution. The mean solution converges exponentially while the gPC reconstruction may fail to converge to the right solution due to the Gibbs phenomenon in the random space. Numerical results confirm the accuracy and spectral convergence of the method.^For the last problem a hybrid method based on the spectral method and weighted essentially non-oscillatory (WENO) finite difference method is proposed to solve the unsteady transonic equations.

Advances in Numerical Methods for Hyperbolic Balance Laws and Related Problems

Advances in Numerical Methods for Hyperbolic Balance Laws and Related Problems

Author: Giacomo Albi

Publisher: Springer Nature

Published: 2023-06-02

Total Pages: 241

ISBN-13: 3031298756

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A broad range of phenomena in science and technology can be described by non-linear partial differential equations characterized by systems of conservation laws with source terms. Well known examples are hyperbolic systems with source terms, kinetic equations, and convection-reaction-diffusion equations. This book collects research advances in numerical methods for hyperbolic balance laws and kinetic equations together with related modelling aspects. All the contributions are based on the talks of the speakers of the Young Researchers’ Conference “Numerical Aspects of Hyperbolic Balance Laws and Related Problems”, hosted at the University of Verona, Italy, in December 2021.


Book Synopsis Advances in Numerical Methods for Hyperbolic Balance Laws and Related Problems by : Giacomo Albi

Download or read book Advances in Numerical Methods for Hyperbolic Balance Laws and Related Problems written by Giacomo Albi and published by Springer Nature. This book was released on 2023-06-02 with total page 241 pages. Available in PDF, EPUB and Kindle. Book excerpt: A broad range of phenomena in science and technology can be described by non-linear partial differential equations characterized by systems of conservation laws with source terms. Well known examples are hyperbolic systems with source terms, kinetic equations, and convection-reaction-diffusion equations. This book collects research advances in numerical methods for hyperbolic balance laws and kinetic equations together with related modelling aspects. All the contributions are based on the talks of the speakers of the Young Researchers’ Conference “Numerical Aspects of Hyperbolic Balance Laws and Related Problems”, hosted at the University of Verona, Italy, in December 2021.

Partial Differential Equations of Hyperbolic Type and Applications

Partial Differential Equations of Hyperbolic Type and Applications

Author: Giuseppe Geymonat

Publisher: World Scientific

Published: 1987

Total Pages: 194

ISBN-13: 9789971502058

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This book introduces the general aspects of hyperbolic conservation laws and their numerical approximation using some of the most modern tools: spectral methods, unstructured meshes and ?-formulation. The applications of these methods are found in some significant examples such as the Euler equations. This book, a collection of articles by the best authors in the field, exposes the reader to the frontier of the research and many open problems.


Book Synopsis Partial Differential Equations of Hyperbolic Type and Applications by : Giuseppe Geymonat

Download or read book Partial Differential Equations of Hyperbolic Type and Applications written by Giuseppe Geymonat and published by World Scientific. This book was released on 1987 with total page 194 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces the general aspects of hyperbolic conservation laws and their numerical approximation using some of the most modern tools: spectral methods, unstructured meshes and ?-formulation. The applications of these methods are found in some significant examples such as the Euler equations. This book, a collection of articles by the best authors in the field, exposes the reader to the frontier of the research and many open problems.

Finite Volume Methods for Hyperbolic Problems

Finite Volume Methods for Hyperbolic Problems

Author: Randall J. LeVeque

Publisher: Cambridge University Press

Published: 2002-08-26

Total Pages: 582

ISBN-13: 1139434187

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This book, first published in 2002, contains an introduction to hyperbolic partial differential equations and a powerful class of numerical methods for approximating their solution, including both linear problems and nonlinear conservation laws. These equations describe a wide range of wave propagation and transport phenomena arising in nearly every scientific and engineering discipline. Several applications are described in a self-contained manner, along with much of the mathematical theory of hyperbolic problems. High-resolution versions of Godunov's method are developed, in which Riemann problems are solved to determine the local wave structure and limiters are then applied to eliminate numerical oscillations. These methods were originally designed to capture shock waves accurately, but are also useful tools for studying linear wave-propagation problems, particularly in heterogenous material. The methods studied are implemented in the CLAWPACK software package and source code for all the examples presented can be found on the web, along with animations of many of the simulations. This provides an excellent learning environment for understanding wave propagation phenomena and finite volume methods.


Book Synopsis Finite Volume Methods for Hyperbolic Problems by : Randall J. LeVeque

Download or read book Finite Volume Methods for Hyperbolic Problems written by Randall J. LeVeque and published by Cambridge University Press. This book was released on 2002-08-26 with total page 582 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book, first published in 2002, contains an introduction to hyperbolic partial differential equations and a powerful class of numerical methods for approximating their solution, including both linear problems and nonlinear conservation laws. These equations describe a wide range of wave propagation and transport phenomena arising in nearly every scientific and engineering discipline. Several applications are described in a self-contained manner, along with much of the mathematical theory of hyperbolic problems. High-resolution versions of Godunov's method are developed, in which Riemann problems are solved to determine the local wave structure and limiters are then applied to eliminate numerical oscillations. These methods were originally designed to capture shock waves accurately, but are also useful tools for studying linear wave-propagation problems, particularly in heterogenous material. The methods studied are implemented in the CLAWPACK software package and source code for all the examples presented can be found on the web, along with animations of many of the simulations. This provides an excellent learning environment for understanding wave propagation phenomena and finite volume methods.

Hyperbolic Partial Differential Equations

Hyperbolic Partial Differential Equations

Author: Matthew Witten

Publisher: Elsevier

Published: 2014-05-17

Total Pages: 255

ISBN-13: 1483155633

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Hyperbolic Partial Differential Equations, Volume 1: Population, Reactors, Tides and Waves: Theory and Applications covers three general areas of hyperbolic partial differential equation applications. These areas include problems related to the McKendrick/Von Foerster population equations, other hyperbolic form equations, and the numerical solution. This text is composed of 15 chapters and begins with surveys of age specific population interactions, populations models of diffusion, nonlinear age dependent population growth with harvesting, local and global stability for the nonlinear renewal equation in the Von Foerster model, and nonlinear age-dependent population dynamics. The next chapters deal with various applications of hyperbolic partial differential equations to such areas as age-structured fish populations, density dependent growth in a cell colony, boll-weevil-cotton crop modeling, age dependent predation and cannibalism, parasite populations, growth of microorganisms, and stochastic perturbations in the Von Foerster model. These topics are followed by discussions of bifurcation of time periodic solutions of the McKendrick equation; the periodic solution of nonlinear hyperbolic problems; and semigroup theory as applied to nonlinear age dependent population dynamics. Other chapters explore the stability of biochemical reaction tanks, an ADI model for the Laplace tidal equations, the Carleman equation, the nonequilibrium behavior of solids that transport heat by second sound, and the nonlinear hyperbolic partial differential equations and dynamic programming. The final chapters highlight two explicitly numerical applications: a predictor-convex corrector method and the Galerkin approximation in hyperbolic partial differential equations. This book will prove useful to practicing engineers, population researchers, physicists, and mathematicians.


Book Synopsis Hyperbolic Partial Differential Equations by : Matthew Witten

Download or read book Hyperbolic Partial Differential Equations written by Matthew Witten and published by Elsevier. This book was released on 2014-05-17 with total page 255 pages. Available in PDF, EPUB and Kindle. Book excerpt: Hyperbolic Partial Differential Equations, Volume 1: Population, Reactors, Tides and Waves: Theory and Applications covers three general areas of hyperbolic partial differential equation applications. These areas include problems related to the McKendrick/Von Foerster population equations, other hyperbolic form equations, and the numerical solution. This text is composed of 15 chapters and begins with surveys of age specific population interactions, populations models of diffusion, nonlinear age dependent population growth with harvesting, local and global stability for the nonlinear renewal equation in the Von Foerster model, and nonlinear age-dependent population dynamics. The next chapters deal with various applications of hyperbolic partial differential equations to such areas as age-structured fish populations, density dependent growth in a cell colony, boll-weevil-cotton crop modeling, age dependent predation and cannibalism, parasite populations, growth of microorganisms, and stochastic perturbations in the Von Foerster model. These topics are followed by discussions of bifurcation of time periodic solutions of the McKendrick equation; the periodic solution of nonlinear hyperbolic problems; and semigroup theory as applied to nonlinear age dependent population dynamics. Other chapters explore the stability of biochemical reaction tanks, an ADI model for the Laplace tidal equations, the Carleman equation, the nonequilibrium behavior of solids that transport heat by second sound, and the nonlinear hyperbolic partial differential equations and dynamic programming. The final chapters highlight two explicitly numerical applications: a predictor-convex corrector method and the Galerkin approximation in hyperbolic partial differential equations. This book will prove useful to practicing engineers, population researchers, physicists, and mathematicians.

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2020+1

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2020+1

Author: Jens M. Melenk

Publisher: Springer Nature

Published: 2023-06-30

Total Pages: 571

ISBN-13: 3031204328

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The volume features high-quality papers based on the presentations at the ICOSAHOM 2020+1 on spectral and high order methods. The carefully reviewed articles cover state of the art topics in high order discretizations of partial differential equations. The volume presents a wide range of topics including the design and analysis of high order methods, the development of fast solvers on modern computer architecture, and the application of these methods in fluid and structural mechanics computations.


Book Synopsis Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2020+1 by : Jens M. Melenk

Download or read book Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2020+1 written by Jens M. Melenk and published by Springer Nature. This book was released on 2023-06-30 with total page 571 pages. Available in PDF, EPUB and Kindle. Book excerpt: The volume features high-quality papers based on the presentations at the ICOSAHOM 2020+1 on spectral and high order methods. The carefully reviewed articles cover state of the art topics in high order discretizations of partial differential equations. The volume presents a wide range of topics including the design and analysis of high order methods, the development of fast solvers on modern computer architecture, and the application of these methods in fluid and structural mechanics computations.

Theory, Numerics and Applications of Hyperbolic Problems II

Theory, Numerics and Applications of Hyperbolic Problems II

Author: Christian Klingenberg

Publisher: Springer

Published: 2018-06-27

Total Pages: 714

ISBN-13: 3319915487

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The second of two volumes, this edited proceedings book features research presented at the XVI International Conference on Hyperbolic Problems held in Aachen, Germany in summer 2016. It focuses on the theoretical, applied, and computational aspects of hyperbolic partial differential equations (systems of hyperbolic conservation laws, wave equations, etc.) and of related mathematical models (PDEs of mixed type, kinetic equations, nonlocal or/and discrete models) found in the field of applied sciences.


Book Synopsis Theory, Numerics and Applications of Hyperbolic Problems II by : Christian Klingenberg

Download or read book Theory, Numerics and Applications of Hyperbolic Problems II written by Christian Klingenberg and published by Springer. This book was released on 2018-06-27 with total page 714 pages. Available in PDF, EPUB and Kindle. Book excerpt: The second of two volumes, this edited proceedings book features research presented at the XVI International Conference on Hyperbolic Problems held in Aachen, Germany in summer 2016. It focuses on the theoretical, applied, and computational aspects of hyperbolic partial differential equations (systems of hyperbolic conservation laws, wave equations, etc.) and of related mathematical models (PDEs of mixed type, kinetic equations, nonlocal or/and discrete models) found in the field of applied sciences.