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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Bibliographic Details
Main Authors: Gottlieb, Sigal, Ketcheson, David, Shu, Chi-Wang
Format: eBook Book
Language:English
Published: Singapore World Scientific Publishing Co. Pte. Ltd 2011
World Scientific
World Scientific Publishing Company
WORLD SCIENTIFIC
WSPC
Edition:1
Subjects:
Differential equations
Differential equations > Numerical solutions
Numerical & Computational Mathematics
Runge-Kutta formulas
Stability
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
ISBN:9814289264, 9789814289269
Online Access:Get full text
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Table of Contents:
  • 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