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The focus of this monograph is the development of space-time adaptive methods to solve the convection/reaction dominated non-stationary semi-linear advection diffusion reaction (ADR) equations with internal/boundary layers in an accurate and efficient way. After introducing the ADR equations and discontinuous Galerkin discretization, robust residual-based a posteriori error estimators in space and time are derived. The elliptic reconstruction technique is then utilized to derive the a posteriori error bounds for the fully discrete system and to obtain optimal orders of convergence.As coupled surface and subsurface flow over large space and time scales is described by (ADR) equation the methods described in this book are of high importance in many areas of Geosciences including oil and gas recovery, groundwater contamination and sustainable use of groundwater resources, storing greenhouse gases or radioactive waste in the subsurface.
Book Synopsis Adaptive Discontinuous Galerkin Methods for Non-linear Reactive Flows by : Murat Uzunca
Download or read book Adaptive Discontinuous Galerkin Methods for Non-linear Reactive Flows written by Murat Uzunca and published by Birkhäuser. This book was released on 2016-05-17 with total page 111 pages. Available in PDF, EPUB and Kindle. Book excerpt: The focus of this monograph is the development of space-time adaptive methods to solve the convection/reaction dominated non-stationary semi-linear advection diffusion reaction (ADR) equations with internal/boundary layers in an accurate and efficient way. After introducing the ADR equations and discontinuous Galerkin discretization, robust residual-based a posteriori error estimators in space and time are derived. The elliptic reconstruction technique is then utilized to derive the a posteriori error bounds for the fully discrete system and to obtain optimal orders of convergence.As coupled surface and subsurface flow over large space and time scales is described by (ADR) equation the methods described in this book are of high importance in many areas of Geosciences including oil and gas recovery, groundwater contamination and sustainable use of groundwater resources, storing greenhouse gases or radioactive waste in the subsurface.
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 up-to-date 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 high-order shock capturing techniques, discontinuous Galerkin methods, adaptive techniques based upon a-posteriori error analysis.
Book Synopsis Advanced Numerical Approximation of Nonlinear Hyperbolic Equations by : B. Cockburn
Download or read book Advanced Numerical Approximation of Nonlinear Hyperbolic Equations written by B. Cockburn and published by Springer. This book was released on 2006-11-14 with total page 446 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 up-to-date 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 high-order shock capturing techniques, discontinuous Galerkin methods, adaptive techniques based upon a-posteriori error analysis.
Ground water reactive transport models are useful to assess and quantify contaminant precipitation, absorption and migration in subsurface media. Many ground water reactive transport models available today are characterized by varying complexities, strengths, and weaknesses. Selecting accurate, efficient models can be a challenging task. This ebook addresses the needs, issues and challenges relevant to selecting a ground water reactive transport model to evaluate natural attenuation and alternative remediation schemes. It should serve as a handy guide for water resource managers seeking to ach.
Book Synopsis Ground Water Reactive Transport Model: Cover Page; 03 REVISED eBooks End User License Agreement-Website; 04 Contents; 05 Foreword_czheng; 06 Preface; 07 Contributors; 08 Chapter 1_Yeh et al_HYDROGEOCHEMA; 09 Chapter 2_Wheeler et al_IPARS-FINAL; 10 Chapter 3_Xu et al-revised-_TOUGHREACT; 11 Chapter 4_Clement et al_RT3D; 12 Chapter 5_White et al_STOMP-ECKEChem; 13 Chapter 6_Hammond et al_PFLOTRAN; 14 Chapter 7_ Samper et al_CORE2D V4; 15 Chapter 8_ Mayer et al_MIN3P; 16 Chapter 9_ Hao et al_NUFT; 17 Index by : Fan Zhang
Download or read book Ground Water Reactive Transport Model: Cover Page; 03 REVISED eBooks End User License Agreement-Website; 04 Contents; 05 Foreword_czheng; 06 Preface; 07 Contributors; 08 Chapter 1_Yeh et al_HYDROGEOCHEMA; 09 Chapter 2_Wheeler et al_IPARS-FINAL; 10 Chapter 3_Xu et al-revised-_TOUGHREACT; 11 Chapter 4_Clement et al_RT3D; 12 Chapter 5_White et al_STOMP-ECKEChem; 13 Chapter 6_Hammond et al_PFLOTRAN; 14 Chapter 7_ Samper et al_CORE2D V4; 15 Chapter 8_ Mayer et al_MIN3P; 16 Chapter 9_ Hao et al_NUFT; 17 Index written by Fan Zhang and published by Bentham Science Publishers. This book was released on 2012 with total page 254 pages. Available in PDF, EPUB and Kindle. Book excerpt: Ground water reactive transport models are useful to assess and quantify contaminant precipitation, absorption and migration in subsurface media. Many ground water reactive transport models available today are characterized by varying complexities, strengths, and weaknesses. Selecting accurate, efficient models can be a challenging task. This ebook addresses the needs, issues and challenges relevant to selecting a ground water reactive transport model to evaluate natural attenuation and alternative remediation schemes. It should serve as a handy guide for water resource managers seeking to ach.
An annual volume presenting substantive survey articles in numerical analysis and scientific computing.
Book Synopsis Acta Numerica 2001: Volume 10 by : Arieh Iserles
Download or read book Acta Numerica 2001: Volume 10 written by Arieh Iserles and published by Cambridge University Press. This book was released on 2001-08-23 with total page 570 pages. Available in PDF, EPUB and Kindle. Book excerpt: An annual volume presenting substantive survey articles in numerical analysis and scientific computing.
This book is the second volume of proceedings of the 8th conference on "Finite Volumes for Complex Applications" (Lille, June 2017). It includes reviewed contributions reporting successful applications in the fields of fluid dynamics, computational geosciences, structural analysis, nuclear physics, semiconductor theory and other topics. The finite volume method in its various forms is a space discretization technique for partial differential equations based on the fundamental physical principle of conservation, and recent decades have brought significant advances in the theoretical understanding of the method. Many finite volume methods preserve further qualitative or asymptotic properties, including maximum principles, dissipativity, monotone decay of free energy, and asymptotic stability. Due to these properties, finite volume methods belong to the wider class of compatible discretization methods, which preserve qualitative properties of continuous problems at the discrete l evel. This structural approach to the discretization of partial differential equations becomes particularly important for multiphysics and multiscale applications. The book is useful for researchers, PhD and master’s level students in numerical analysis, scientific computing and related fields such as partial differential equations, as well as for engineers working in numerical modeling and simulations.
Book Synopsis Finite Volumes for Complex Applications VIII - Hyperbolic, Elliptic and Parabolic Problems by : Clément Cancès
Download or read book Finite Volumes for Complex Applications VIII - Hyperbolic, Elliptic and Parabolic Problems written by Clément Cancès and published by Springer. This book was released on 2017-05-22 with total page 559 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is the second volume of proceedings of the 8th conference on "Finite Volumes for Complex Applications" (Lille, June 2017). It includes reviewed contributions reporting successful applications in the fields of fluid dynamics, computational geosciences, structural analysis, nuclear physics, semiconductor theory and other topics. The finite volume method in its various forms is a space discretization technique for partial differential equations based on the fundamental physical principle of conservation, and recent decades have brought significant advances in the theoretical understanding of the method. Many finite volume methods preserve further qualitative or asymptotic properties, including maximum principles, dissipativity, monotone decay of free energy, and asymptotic stability. Due to these properties, finite volume methods belong to the wider class of compatible discretization methods, which preserve qualitative properties of continuous problems at the discrete l evel. This structural approach to the discretization of partial differential equations becomes particularly important for multiphysics and multiscale applications. The book is useful for researchers, PhD and master’s level students in numerical analysis, scientific computing and related fields such as partial differential equations, as well as for engineers working in numerical modeling and simulations.
As computational research has grown, simulation has become a standard tool in many fields of academic and industrial areas. For example, computational fluid dynamics (CFD) tools in aerospace and research facilities are widely used to evaluate the aerodynamic performance of aircraft or wings. Weather forecasts are highly dependent on numerical weather prediction (NWP) model. However, it is still difficult to simulate the complex physical phenomena of a wide range of length and time scales with modern computational resources. In this study, we develop a robust, efficient and high-order accurate numerical methods and techniques to tackle the challenges. First, we use high-order spatial discretization using (hybridized) discontinuous Galerkin (DG) methods. The DG method combines the advantages of finite volume and finite element methods. As such, it is well-suited to problems with large gradients including shocks and with complex geometries, and large-scale simulations. However, DG typically has many degrees-of-freedoms. To mitigate the expense, we use hybridized DG (HDG) method that introduces new “trace unknowns” on the mesh skeleton (mortar interfaces) to eliminate the local “volume unknowns” with static condensation procedure and reduces globally coupled system when implicit time-stepping is required. Also, since the information between the elements is exchanged through the mesh skeleton, the mortar interfaces can be used as a glue to couple multi-phase regions, e.g., solid and fluid regions, or non-matching grids, e.g., a rotating mesh and a stationary mesh. That is the HDG method provides an efficient and flexible coupling environment compared to standard DG methods. Second, we develop an HDG-DG IMEX scheme for an efficient time integrating scheme. The idea is to divide the governing equations into stiff and nonstiff parts, implicitly treat the former with HDG methods, and explicitly treat the latter with DG methods. The HDG-DG IMEX scheme facilitates high-order temporal and spatial solutions, avoiding too small a time step. Numerical results show that the HDG-DG IMEX scheme is comparable to an explicit Runge-Kutta DG scheme in terms of accuracy while allowing for much larger timestep sizes. We also numerically observe that IMEX HDG-DG scheme can be used as a tool to suppress the high-frequency modes such as acoustic waves or fast gravity waves in atmospheric or ocean models. In short, IMEX HDG-DG methods are attractive for applications in which a fast and stable solution is important while permitting inaccurate processing of the fast modes. Third, we also develop an EXPONENTIAL DG scheme for an efficient time integrators. Similar to the IMEX method, the governing equations are separated into linear and nonlinear parts, then the two parts are spatially discretized with DG methods. Next, we analytically integrate the linear term and approximate the nonlinear term with respect to time. This method accurately handles the fast wave modes in the linear operator. To efficiently evaluate a matrix exponential, we employ the cutting-edge adaptive Krylov subspace method. Finally, we develop a sliding-mesh interface by combining nonconforming treatment and the arbitrary Lagrangian-Eulerian (ALE) scheme for simulating rotating flows, which are important to estimate the characteristics of a rotating wind turbine or understanding vortical structures shown in atmospheric or astronomical phenomena. To integrate the rotating motion of the domain, we use the ALE formulation to map the governing equation to the stationary reference domain and introduce mortar interfaces between the stationary mesh and the rotating mesh. The mortar structure on the sliding interface changes dynamically as the mesh rotates
Book Synopsis High-order (hybridized) Discontinuous Galerkin Method for Geophysical Flows by : Shinhoo Kang
Download or read book High-order (hybridized) Discontinuous Galerkin Method for Geophysical Flows written by Shinhoo Kang and published by . This book was released on 2019 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: As computational research has grown, simulation has become a standard tool in many fields of academic and industrial areas. For example, computational fluid dynamics (CFD) tools in aerospace and research facilities are widely used to evaluate the aerodynamic performance of aircraft or wings. Weather forecasts are highly dependent on numerical weather prediction (NWP) model. However, it is still difficult to simulate the complex physical phenomena of a wide range of length and time scales with modern computational resources. In this study, we develop a robust, efficient and high-order accurate numerical methods and techniques to tackle the challenges. First, we use high-order spatial discretization using (hybridized) discontinuous Galerkin (DG) methods. The DG method combines the advantages of finite volume and finite element methods. As such, it is well-suited to problems with large gradients including shocks and with complex geometries, and large-scale simulations. However, DG typically has many degrees-of-freedoms. To mitigate the expense, we use hybridized DG (HDG) method that introduces new “trace unknowns” on the mesh skeleton (mortar interfaces) to eliminate the local “volume unknowns” with static condensation procedure and reduces globally coupled system when implicit time-stepping is required. Also, since the information between the elements is exchanged through the mesh skeleton, the mortar interfaces can be used as a glue to couple multi-phase regions, e.g., solid and fluid regions, or non-matching grids, e.g., a rotating mesh and a stationary mesh. That is the HDG method provides an efficient and flexible coupling environment compared to standard DG methods. Second, we develop an HDG-DG IMEX scheme for an efficient time integrating scheme. The idea is to divide the governing equations into stiff and nonstiff parts, implicitly treat the former with HDG methods, and explicitly treat the latter with DG methods. The HDG-DG IMEX scheme facilitates high-order temporal and spatial solutions, avoiding too small a time step. Numerical results show that the HDG-DG IMEX scheme is comparable to an explicit Runge-Kutta DG scheme in terms of accuracy while allowing for much larger timestep sizes. We also numerically observe that IMEX HDG-DG scheme can be used as a tool to suppress the high-frequency modes such as acoustic waves or fast gravity waves in atmospheric or ocean models. In short, IMEX HDG-DG methods are attractive for applications in which a fast and stable solution is important while permitting inaccurate processing of the fast modes. Third, we also develop an EXPONENTIAL DG scheme for an efficient time integrators. Similar to the IMEX method, the governing equations are separated into linear and nonlinear parts, then the two parts are spatially discretized with DG methods. Next, we analytically integrate the linear term and approximate the nonlinear term with respect to time. This method accurately handles the fast wave modes in the linear operator. To efficiently evaluate a matrix exponential, we employ the cutting-edge adaptive Krylov subspace method. Finally, we develop a sliding-mesh interface by combining nonconforming treatment and the arbitrary Lagrangian-Eulerian (ALE) scheme for simulating rotating flows, which are important to estimate the characteristics of a rotating wind turbine or understanding vortical structures shown in atmospheric or astronomical phenomena. To integrate the rotating motion of the domain, we use the ALE formulation to map the governing equation to the stationary reference domain and introduce mortar interfaces between the stationary mesh and the rotating mesh. The mortar structure on the sliding interface changes dynamically as the mesh rotates
The use of high-order discontinuous Galerkin (DG) discretizations has become more widespread over the last decade for solving convection-dominated computational fluid dynamics problems. The appeal of these methods relates to their favorable asymptotic accuracy properties, combined with compact stencils and favorable scalability properties on parallel computing architectures. This work covers advances in several areas of high-order DG discretizations, including the development of implicit solvers, discrete adjoint methods for shape optimization, and output-based error estimation and mesh and time-step adaptation. For time-dependent problems, high-order implicit time-integration schemes are considered exclusively to avoid the stability restrictions of explicit methods, with particular emphasis on balancing spatial and temporal accuracy of the overall approach. In order to make the high-order schemes competitive, efficient solution techniques consisting of a p -multigrid approach driven by element Jacobi smoothers are investigated and developed to accelerate convergence of the non-linear systems, in which the results demonstrate h independent convergence rates, while remaining relatively insensitive to time-step sizes. A framework based on discrete adjoint sensitivity analysis has also been developed for applications in shape optimization and goal-oriented error estimation. An adaptive discontinuous Galerkin algorithm driven by an adjoint-based error estimation procedure has been developed, which incorporates both h-, p- and combined hp -adaptive schemes, for producing accurate simulations at optimal cost in the objective functional of interest. Current results show superior performance of these adaptive schemes over uniform mesh refinement methods, as well as the potential of the hp refinement approach to capture strong shocks without limiters. Finally, the adjoint-based error estimation strategy is successfully extended to unsteady flow problems, where the time-dependent flow solution is solved in a forward manner in time but the corresponding unsteady adjoint solution is evaluated as a backward time integration. Results demonstrate that this methodology provides accurate global temporal error prediction, and may be employed to drive an adaptive time-step refinement strategy for improving the accuracy of specified time-dependent functionals of interest.
Book Synopsis Techniques for High-order Adaptive Discontinuous Galerkin Discretizations in Fluid Dynamics by : Li Wang
Download or read book Techniques for High-order Adaptive Discontinuous Galerkin Discretizations in Fluid Dynamics written by Li Wang and published by . This book was released on 2009 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt: The use of high-order discontinuous Galerkin (DG) discretizations has become more widespread over the last decade for solving convection-dominated computational fluid dynamics problems. The appeal of these methods relates to their favorable asymptotic accuracy properties, combined with compact stencils and favorable scalability properties on parallel computing architectures. This work covers advances in several areas of high-order DG discretizations, including the development of implicit solvers, discrete adjoint methods for shape optimization, and output-based error estimation and mesh and time-step adaptation. For time-dependent problems, high-order implicit time-integration schemes are considered exclusively to avoid the stability restrictions of explicit methods, with particular emphasis on balancing spatial and temporal accuracy of the overall approach. In order to make the high-order schemes competitive, efficient solution techniques consisting of a p -multigrid approach driven by element Jacobi smoothers are investigated and developed to accelerate convergence of the non-linear systems, in which the results demonstrate h independent convergence rates, while remaining relatively insensitive to time-step sizes. A framework based on discrete adjoint sensitivity analysis has also been developed for applications in shape optimization and goal-oriented error estimation. An adaptive discontinuous Galerkin algorithm driven by an adjoint-based error estimation procedure has been developed, which incorporates both h-, p- and combined hp -adaptive schemes, for producing accurate simulations at optimal cost in the objective functional of interest. Current results show superior performance of these adaptive schemes over uniform mesh refinement methods, as well as the potential of the hp refinement approach to capture strong shocks without limiters. Finally, the adjoint-based error estimation strategy is successfully extended to unsteady flow problems, where the time-dependent flow solution is solved in a forward manner in time but the corresponding unsteady adjoint solution is evaluated as a backward time integration. Results demonstrate that this methodology provides accurate global temporal error prediction, and may be employed to drive an adaptive time-step refinement strategy for improving the accuracy of specified time-dependent functionals of interest.
The subject of the book is the mathematical theory of the discontinuous Galerkin method (DGM), which is a relatively new technique for the numerical solution of partial differential equations. The book is concerned with the DGM developed for elliptic and parabolic equations and its applications to the numerical simulation of compressible flow. It deals with the theoretical as well as practical aspects of the DGM and treats the basic concepts and ideas of the DGM, as well as the latest significant findings and achievements in this area. The main benefit for readers and the book’s uniqueness lie in the fact that it is sufficiently detailed, extensive and mathematically precise, while at the same time providing a comprehensible guide through a wide spectrum of discontinuous Galerkin techniques and a survey of the latest efficient, accurate and robust discontinuous Galerkin schemes for the solution of compressible flow.
Book Synopsis Discontinuous Galerkin Method by : Vít Dolejší
Download or read book Discontinuous Galerkin Method written by Vít Dolejší and published by Springer. This book was released on 2015-07-17 with total page 575 pages. Available in PDF, EPUB and Kindle. Book excerpt: The subject of the book is the mathematical theory of the discontinuous Galerkin method (DGM), which is a relatively new technique for the numerical solution of partial differential equations. The book is concerned with the DGM developed for elliptic and parabolic equations and its applications to the numerical simulation of compressible flow. It deals with the theoretical as well as practical aspects of the DGM and treats the basic concepts and ideas of the DGM, as well as the latest significant findings and achievements in this area. The main benefit for readers and the book’s uniqueness lie in the fact that it is sufficiently detailed, extensive and mathematically precise, while at the same time providing a comprehensible guide through a wide spectrum of discontinuous Galerkin techniques and a survey of the latest efficient, accurate and robust discontinuous Galerkin schemes for the solution of compressible flow.
There has been rapid development in the area of adaptive computation over the past decade. The International Conference on Recent Advances in Adaptive Computation was held at Zhejiang University (Hangzhou, China) to explore these new directions. The conference brought together specialists to discuss modern theories and practical applications of adaptive methods. This volume contains articles reflecting the invited talks given by leading mathematicians at the conference. It is suitable for graduate students and researchers interested in methods of computation.
Book Synopsis Recent Advances in Adaptive Computation by : Zhongci Shi
Download or read book Recent Advances in Adaptive Computation written by Zhongci Shi and published by American Mathematical Soc.. This book was released on 2005 with total page 394 pages. Available in PDF, EPUB and Kindle. Book excerpt: There has been rapid development in the area of adaptive computation over the past decade. The International Conference on Recent Advances in Adaptive Computation was held at Zhejiang University (Hangzhou, China) to explore these new directions. The conference brought together specialists to discuss modern theories and practical applications of adaptive methods. This volume contains articles reflecting the invited talks given by leading mathematicians at the conference. It is suitable for graduate students and researchers interested in methods of computation.
The articles in this volume summarize the research results obtained in the former SFB 359 "Reactive Flow, Diffusion and Transport" which has been supported by the DFG over the period 1993-2004. The main subjects are physical-chemical processes sharing the difficulty of interacting diffusion, transport and reaction which cannot be considered separately. The modeling and simulation within this book is accompanied by experiments.
Book Synopsis Reactive Flows, Diffusion and Transport by : Willi Jäger
Download or read book Reactive Flows, Diffusion and Transport written by Willi Jäger and published by Springer Science & Business Media. This book was released on 2007-05-31 with total page 659 pages. Available in PDF, EPUB and Kindle. Book excerpt: The articles in this volume summarize the research results obtained in the former SFB 359 "Reactive Flow, Diffusion and Transport" which has been supported by the DFG over the period 1993-2004. The main subjects are physical-chemical processes sharing the difficulty of interacting diffusion, transport and reaction which cannot be considered separately. The modeling and simulation within this book is accompanied by experiments.
