Strong stability preserving Runge-Kutta and multistep time discretizations
This book captures the state-of-the-art in the field of Strong Stability Preserving (SSP) time stepping methods, which have significant advantages for the time evolution of partial differential equations describing a wide range of physical phenomena. This comprehensive book describes the development...
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| Médium: | E-kniha Kniha |
| Jazyk: | angličtina |
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Singapore
World Scientific Publishing Co. Pte. Ltd
2011
World Scientific World Scientific Publishing Company WORLD SCIENTIFIC WSPC |
| Vydání: | 1 |
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| ISBN: | 9814289264, 9789814289269 |
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| Abstract | This book captures the state-of-the-art in the field of Strong Stability Preserving (SSP) time stepping methods, which have significant advantages for the time evolution of partial differential equations describing a wide range of physical phenomena. This comprehensive book describes the development of SSP methods, explains the types of problems which require the use of these methods and demonstrates the efficiency of these methods using a variety of numerical examples. Another valuable feature of this book is that it collects the most useful SSP methods, both explicit and implicit, and presents the other properties of these methods which make them desirable (such as low storage, small error coefficients, large linear stability domains). This book is valuable for both researchers studying the field of time-discretizations for PDEs, and the users of such methods. |
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| AbstractList | This book captures the state-of-the-art in the field of Strong Stability Preserving (SSP) time stepping methods, which have significant advantages for the time evolution of partial differential equations describing a wide range of physical phenomena. This comprehensive book describes the development of SSP methods, explains the types of problems which require the use of these methods and demonstrates the efficiency of these methods using a variety of numerical examples. Another valuable feature of this book is that it collects the most useful SSP methods, both explicit and implicit, and presents the other properties of these methods which make them desirable (such as low storage, small error coefficients, large linear stability domains). This book is valuable for both researchers studying the field of time-discretizations for PDEs, and the users of such methods. |
| Author | Shu, Chi-Wang Gottlieb, Sigal Ketcheson, David |
| Author_xml | – sequence: 1 fullname: Gottlieb, Sigal – sequence: 2 fullname: Gottlieb, Sigal – sequence: 3 fullname: Ketcheson, David – sequence: 4 fullname: Shu, Chi-Wang |
| BackLink | https://cir.nii.ac.jp/crid/1130000794306373376$$DView record in CiNii |
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| DOI | 10.1142/7498 |
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| Discipline | Applied Sciences Mathematics |
| EISBN | 9789814289276 9814289272 9789814466479 9814466476 |
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| Keywords | Strong Stability Preserving SSP Time Stepping Methods Time Discretizations Runge-Kutta Methods Implicit Explicit Methods Multi-Step Multi-Stage Methods Hyperbolic PDEs Multi-Step Methods General Linear Methods Downwinding |
| LCCN | 2010048026 |
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| Language | English |
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| Notes | Includes bibliographical references (p. 167-173) and index |
| OCLC | 733048132 |
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| PageCount | 189 |
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| PublicationDecade | 2010 |
| PublicationPlace | Singapore |
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| Publisher | World Scientific Publishing Co. Pte. Ltd World Scientific World Scientific Publishing Company WORLD SCIENTIFIC WSPC |
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| Snippet | This book captures the state-of-the-art in the field of Strong Stability Preserving (SSP) time stepping methods, which have significant advantages for the time... |
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| SubjectTerms | Differential equations Differential equations -- Numerical solutions Numerical & Computational Mathematics Runge-Kutta formulas Stability |
| SubjectTermsDisplay | Differential equations Runge-Kutta formulas Stability |
| TableOfContents | Strong stability preserving Runge-Kutta and multistep time discretizations -- Preface -- Contents -- Chapter 1: Overview: The Development of SSP Methods -- Chapter 2: Strong Stability Preserving Explicit Runge-Kutta Methods -- Chapter 3: The SSP Coefficient for Runge-Kutta Methods -- Chapter 4: SSP Runge-Kutta Methods for Linear Constant Coefficient Problems -- Chapter 5: Bounds and Barriers for SSP Runge-Kutta Methods -- Chapter 6: Low Storage Optimal Explicit SSP Runge-Kutta Methods -- Chapter 7: Optimal Implicit SSP Runge-Kutta Methods -- Chapter 8: SSP Properties of Linear Multistep Methods -- Chapter 9: SSP Properties of Multistep Multi-Stage Methods -- Chapter 10: Downwinding -- Chapter 11: Applications -- Bibliography -- Index Intro -- Contents -- Preface -- 1. Overview: The Development of SSP Methods -- 2. Strong Stability Preserving Explicit Runge-Kutta Methods -- 2.1 Overview -- 2.2 Motivation -- 2.3 SSP methods as convex combinations of Euler's method: the Shu-Osher formulation -- 2.4 Some optimal SSP Runge{Kutta methods -- 2.4.1 A second order method -- 2.4.2 A third order method -- 2.4.3 A fourth order method -- 3. The SSP Coe cient for Runge-Kutta Methods -- 3.1 The modi ed Shu-Osher form -- 3.1.1 Vector notation -- 3.2 Unique representations -- 3.2.1 The Butcher form -- 3.2.2 Reducibility of Runge{Kutta methods -- 3.3 The canonical Shu-Osher form -- 3.3.1 Computing the SSP coefficient -- 3.4 Formulating the optimization problem -- 3.5 Necessity of the SSP time step restriction -- 4. SSP Runge-Kutta Methods for Linear Constant Coefficient Problems -- 4.1 The circle condition -- 4.2 An example: the midpoint method -- 4.3 The stability function -- 4.3.1 Formulas for the stability function -- 4.3.2 An alternative form -- 4.3.3 Order conditions on the stability function -- 4.4 Strong stability preservation for linear systems -- 4.5 Absolute monotonicity -- 4.6 Necessity of the time step condition -- 4.7 An optimization algorithm -- 4.8 Optimal SSP Runge{Kutta methods for linear problems -- 4.9 Linear constant coe cient operators with time dependent forcing terms -- 5. Bounds and Barriers for SSP Runge-Kutta Methods -- 5.1 Order barriers -- 5.1.1 Stage order -- 5.1.2 Order barrier for explicit Runge-Kutta methods -- 5.1.3 Order barrier for implicit Runge-Kutta methods -- 5.1.4 Order barriers for diagonally implicit and singly implicit methods -- 5.2 Bounds on the SSP coefficient -- 5.2.1 Bounds for explicit Runge-Kutta methods -- 5.2.2 Unconditional strong stability preservation -- 6. Low Storage Optimal Explicit SSP Runge-Kutta Methods 6.1 Low-storage Runge-Kutta algorithms -- 6.1.1 Williamson (2N) methods -- 6.1.2 van der Houwen (2R) methods -- 6.1.3 2S and 2S* methods -- 6.2 Optimal SSP low-storage explicit Runge{Kutta methods -- 6.2.1 Second order methods -- 6.2.2 Third order methods -- 6.2.3 Fourth order methods -- 6.3 Embedded optimal SSP pairs -- 7. Optimal Implicit SSP Runge-Kutta Methods -- 7.1 Barriers, bounds, and bonuses -- 7.2 Optimal second order and third order methods -- 7.3 Optimal fourth through sixth order methods -- 7.4 Coefficients of optimal implicit SSP Runge-Kutta methods -- 7.4.1 Fourth order methods -- 7.4.2 Fifth order methods -- 7.4.3 Sixth order methods -- 8. SSP Properties of Linear Multistep Methods -- 8.1 Bounds and barriers -- 8.1.1 Explicit methods -- 8.1.2 Implicit methods -- 8.2 Explicit SSP multistep methods using few stages -- 8.2.1 Second order methods -- 8.2.2 Third order methods -- 8.2.3 Fourth order methods -- 8.3 Optimal methods of higher order and more steps -- 8.3.1 Explicit methods -- 8.3.2 Implicit methods -- 8.4 Starting methods -- 9. SSP Properties of Multistep Multi-Stage Methods -- 9.1 SSP theory of general linear methods -- 9.2 Two-step Runge{Kutta methods -- 9.2.1 Conditions and barriers for SSP two-step Runge-Kutta methods -- 9.3 Optimal two-step Runge-Kutta methods -- 9.3.1 Formulating the optimization problem -- 9.3.2 Efficient implementation of Type II SSP TSRKs -- 9.3.3 Optimal methods of orders one to four -- 9.3.4 Optimal methods of orders ve to eight -- 9.4 Coefficients of optimal methods -- 9.4.1 Fifth order SSP TSRK methods -- 9.4.2 Sixth order SSP TSRK methods -- 9.4.3 Seventh order SSP TSRK methods -- 9.4.4 Eighth order SSP TSRK methods -- 10. Downwinding -- 10.1 SSP methods with negative ij 's -- 10.2 Explicit SSP Runge-Kutta methods with downwinding -- 10.2.1 Second and third order methods 10.2.2 Fourth order methods -- 10.2.3 A fifth order method -- 10.3 Optimal explicit multistep schemes with downwinding -- 10.4 Application: Deferred correction methods -- 11. Applications -- 11.1 TVD schemes -- 11.2 Maximum principle satisfying schemes and positivity preserving schemes -- 11.3 Coercive approximations -- Bibliography -- Index |
| Title | Strong stability preserving Runge-Kutta and multistep time discretizations |
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