ssp.bib
@article{bellen1994,
author = {A. Bellen and Z. Jackiewicz and M. Zennaro},
journal = {SIAM Journal of Numerical Analysis},
pages = {499-523},
title = {Contractivity of waveform relaxation {R}unge-{K}utta iterations and related limit methods for dissipative systems in the maximum norm},
volume = {31},
year = {1994}
}
@article{bellen1997,
author = {A. Bellen and L. Torelli},
journal = {SIAM Journal of Numerical Analysis},
pages = {528-543},
title = {Unconditional Contractivity in the maximum norm of diagonally split {R}unge-{K}utta methods},
volume = {34},
year = {1997}
}
@article{ferracina2005a,
abstract = {In the literature, on the numerical solution of nonlinear time dependent partial differential equations, much attention has been paid to numerical processes which have the favourable property of being total variation bounded (TVB). A popular approach to guaranteeing the TVB property consists in demanding that the process has the stronger property of being total variation diminishing (TVD).
For Runge-Kutta methods-applied to semi-discrete approximations of partial differential equations-conditions on the time step were established which guarantee the TVD property; see, e.g., [J. Comput. Phys. 77 (1988) 439; Math. Comp. 67 (1998) 73; SIAM Rev. 43 (2001) 89; SIAM J. Numer. Anal. (2002), in press; Higueras, Tech. Report, Universidad Publica de Navarra, 2002; SIAM J. Numer. Anal. 40 (2002) 469]. These conditions were derived under the assumption that the simple explicit Euler time stepping process is TVD.
However, for various important semi-discrete approximations, the Euler process is TVB but not TVD-see, e.g., [Math. Comp. 49 (1987) 105; Math. Comp. 52 (1989) 411]. Accordingly, the above stepsize conditions for RungeKutta methods are not directly relevant to such approximations, and there is a need for stepsize restrictions with a wider range of applications.
In this paper, we propose a general theory yielding stepsize restrictions which cover a larger class of semidiscrete approximations than covered thus far in the literature. In particular, our theory gives stepsize restrictions, for general Runge-Kutta methods, which guarantee total-variation-boundedness in situations where the Euler process is TVB but not TVD. (c) 2004 IMACS. Published by Elsevier B.V. All rights reserved.},
author = {Ferracina, L and Spijker, MN},
doi = {DOI 10.1016/j.apnum.2004.08.024},
journal = {Applied Numerical Mathematics},
keywords = {initial value problem; conservation law; method of lines; Runge-Kutta methods; total variation diminishing; total variation bounded (TVB); Runge-Kutta methods},
pages = {265-279},
title = {Stepsize restrictions for total-variation-boundedness in general Runge-Kutta procedures},
volume = {53},
year = {2005},
}
@article{ferracina2005,
author = {L. Ferracina and M. N. Spijker},
journal = {Mathematics of Computation},
pages = {201-219},
title = {An extension and analysis of the {S}hu-{O}sher representation of {R}unge-{K}utta methods},
volume = {249},
year = {2005}
}
@techreport{ferracina2005b,
author = {L. Ferracina and M. N. Spijker},
institution = {Mathematical Institute, Leiden University},
number = {MI2005-07},
title = {Computing optimal monotonicity-preserving {R}unge-{K}utta methods},
year = {2005}
}
@article{ferracina2004,
author = {L. Ferracina and M. N. Spijker},
journal = {SIAM Journal of Numerical Analysis},
pages = {1073-1093},
title = {Stepsize restrictions for the total-variation-diminishing property in general {R}unge-{K}utta methods},
volume = {42},
year = {2004}
}
@phdthesis{ferracina2005c,
author = {Luca Ferracina},
school = {University of Leiden},
title = {Monotonicity and boundedness in general {R}unge-{K}utta methods},
year = {2005},
}
@article{ferracina2008,
author = {Luca Ferracina and Marc Spjker},
journal = {Applied Numerical Mathematics},
note = {to appear},
title = {Strong Stability of Singly-Diagonally-Implicit {R}unge-{K}utta Methods},
year = {2008}
}
@article{gottlieb2005,
author = {Sigal Gottlieb},
journal = {Journal of Scientific Computing},
pages = {105-127},
title = {On high order strong stability preserving {R}unge-{K}utta and multi step time discretizations},
volume = {25},
year = {2005}
}
@article{gottlieb2006,
author = {Sigal Gottlieb and Steven J. Ruuth},
journal = {Journal of Scientific Computing},
pages = {289-303},
title = {Optimal strong-stability-preserving time-stepping schemes with fast downwind spatial discretizations},
volume = {27},
year = {2006}
}
@article{gottlieb2001,
author = {Sigal Gottlieb and Chi-Wang Shu and Eitan Tadmor},
journal = {SIAM Review},
pages = {89-112},
title = {Strong Stability Preserving High-Order Time Discretization Methods},
volume = {43},
year = {2001}
}
@article{gottlieb2003,
author = {Sigal Gottlieb and Lee-Ad J. Gottlieb},
journal = {Journal of Scientific Computing},
pages = {83-109},
title = {Strong Stability Preserving Properties of {R}unge-{K}utta Time Discretization Methods for Linear Constant Coefficient Operators},
volume = {18},
year = {2003}
}
@article{gottlieb1998,
author = {Sigal Gottlieb and Chi-Wang Shu},
journal = {Mathematics of Computation},
pages = {73-85},
title = {Total Variation Diminishing {R}unge-{K}utta Schemes},
volume = {67},
year = {1998},
}
@article{vandegriend1986,
author = {J. A. van de Griend and J. F. B. M. Kraaijevanger},
journal = {Numerische Mathematik},
pages = {413-424},
title = {Absolute Monotonicity of Rational Functions Occurring in the Numerical Solution of Initial Value Problems},
volume = {49},
year = {1986}
}
@article{hairer1980,
author = {E. Hairer},
journal = {BIT},
pages = {254-256},
title = {Highest possible order of algebraically stable diagonally implicit {R}unge-{K}utta methods},
volume = {20},
year = {1980}
}
@article{higueras2004a,
author = {Higueras, I},
journal = {Journal of Scientific Computing},
keywords = {strong stability preservation; strong stability preservation; monotonicity; Runge-Kutta methods},
pages = {193-223},
title = {On strong stability preserving time discretization methods},
volume = {21},
year = {2004}
}
@article{higueras2005a,
abstract = {Over the last few years a great effort has been made to develop monotone high order explicit Runge-Kutta methods by means of their Shu-Osher representations. In this context, the stepsize restriction to obtain numerical monotonicity is normally computed using the optimal representation. In this paper we extend the Shu-Osher representations for any Runge-Kutta method giving sufficient conditions for monotonicity. We show how optimal Shu-Osher representations can be constructed from the Butcher tableau of a Runge-Kutta method.
The optimum stepsize restriction for monotonicity is given by the radius of absolute monotonicity of the Runge-Kutta method [L. Ferracina and M. N. Spijker, SIAM J. Numer. Anal., 42 (2004), pp. 1073-1093], and hence if this radius is zero, the method is not monotone. In the Shu-Osher representation, methods with zero radius require negative coefficients, and to deal with them, an extra associate problem is considered. In this paper we interpret these schemes as representations of perturbed Runge-Kutta methods. We extend the concept of radius of absolute monotonicity and give sufficient conditions for monotonicity. Optimal representations can be constructed from the Butcher tableau of a perturbed Runge-Kutta method.},
author = {Higueras, I},
doi = {DOI 10.1137/S0036142903427068},
journal = {Siam Journal On Numerical Analysis},
keywords = {Runge-Kutta methods; strong stability preservation; strong stability preservation; absolutely monotonic; radius of absolute monotonicity; CFL coefficient; representations},
pages = {924-948},
title = {Representations of {R}unge-{K}utta methods and strong stability preserving methods},
volume = {43},
year = {2005},
}
@article{higueras2006,
author = {Inmaculada Higueras},
journal = {SIAM J. Numer. Anal.},
pages = {1735-1758},
title = {Strong Stability for Additive {R}unge-{K}utta Methods},
volume = {44},
year = {2006}
}
@article{horvath2005,
author = {Zoltan Horvath},
journal = {Applied Numerical Mathematics},
pages = {341-356},
title = {On the positivity step size threshold of {R}unge-{K}utta methods},
volume = {53},
year = {2005}
}
@article{horvath1998,
author = {Zoltan Horvath},
journal = {Applied Numerical Mathematics},
pages = {309-326},
title = {Positivity of {R}unge-{K}utta and Diagonally Split {R}unge-{K}utta Methods},
volume = {28},
year = {1998}
}
@article{hout1996,
abstract = {In this paper we consider diagonally split Runge-Kutta methods for the numerical solution of initial value problems for ordinary differential equations. This class of numerical methods was recently introduced by Bellen, Jackiewicz, and Zennaro [SIAM J. Numer. Anal., 31 (1994), pp. 499-523], and comprises the well-known class of Runge-Kutta methods. Their results strongly indicate that diagonally split Runge-Kutta methods break the order barrier p less than or equal to 1 for unconditional contractivity in the maximum norm. In this paper we investigate the effect of the requirement of unconditional contractivity in the maximum norm on the accuracy of a diagonally split Runge-Kutta method. Besides the classical order p, we deal with an order of accuracy r which is relevant to the case where the method is applied to dissipative initial value problems that are arbitrarily stiff We show that if a diagonally split Runge-Kutta method is unconditionally contractive in the maximum norm, then it has orders p, r which satisfy p less than or equal to 4, r less than or equal to 1.},
author = {Hout, KJI},
journal = {Siam Journal On Numerical Analysis},
keywords = {ordinary differential equations; dissipative initial value problems; maximum norm; diagonally split Runge-Kutta methods; Runge-Kutta methods; unconditional contractivity; accuracy},
pages = {1125-1134},
title = {A note on unconditional maximum norm contractivity of diagonally split Runge-Kutta methods},
volume = {33},
year = {1996}
}
@article{hundsdorfer2007,
author = {Willem Hundsdorfer and Steven J. Ruuth},
journal = {Journal of Computational Physics},
pages = {201-2042},
title = {IMEX extensions of linear multistep methods with general monotonicity and boundedness properties},
volume = {225},
year = {2007}
}
@article{hundsdorfer2003,
author = {Willem Hundsdorfer and Steven J. Ruuth and Raymond J. Spiteri},
journal = {SIAM Journal of Numerical Analysis},
pages = {605-623},
title = {Monotonicity-Preserving Linear Multistep Methods},
volume = {41},
year = {2003}
}
@article{hundsdorfer2005,
author = {Willem Hundsdorfer and Steven J. Ruuth},
journal = {Mathematics of Computation},
number = {254},
pages = {655-672},
title = {On monotonicity and boundedness properties of linear multistep methods},
volume = {75},
year = {2005},
}
@article{kennedy2000,
author = {Christopher A. Kennedy and Mark H. Carpenter and R. Michael Lewis},
journal = {Applied Numerical Mathematics},
pages = {177-219},
title = {Low-Storage, Explicit {R}unge-{K}utta Schemes for the compressible {N}avier-{S}tokes Equations},
volume = {35},
year = {2000}
}
@mastersthesis{ketcheson2004,
author = {David I. Ketcheson},
note = {B.Sc. Thesis},
school = {Brigham Young University},
title = {An algebraic characterization of strong stability preserving {R}unge-{K}utta schemes},
year = {2004}
}
@article{ketcheson2007b,
author = {David I. Ketcheson},
journal = {SIAM Journal on Scientific Computing},
note = {to appear},
title = {Highly Efficient Strong Stability Preserving {R}unge-{K}utta Methods with Low-Storage Implementations},
year = {2008}
}
@article{ketcheson2005,
author = {David I. Ketcheson and Allen C. Robinson},
journal = {International Journal for Numerical Methods in Fluids},
pages = {271-303},
title = {On the practical importance of the {SSP} property for {R}unge-{K}utta time integrators for some common {G}odunov-type schemes},
volume = {48},
year = {2005}
}
@article{ketcheson2007,
author = {David I. Ketcheson and Colin B. Macdonald and Sigal Gottlieb},
note = {in preparation},
title = {Optimal Implicit Strong Stability Preserving {R}unge-{K}utta Methods},
year = {2007}
}
@article{kraaijevanger1986,
author = {J. F. B. M. Kraaijevanger},
journal = {Numerische Mathematik},
pages = {303-322},
title = {Absolute monotonicity of polynomials occurring in the numerical solution of initial value problems},
volume = {48},
year = {1986}
}
@article{kraaijevanger1991,
author = {J. F. B. M. Kraaijevanger},
journal = {BIT},
pages = {482-528},
title = {Contractivity of {R}unge-{K}utta Methods},
volume = {31},
year = {1991}
}
@article{kubatko2007,
author = {Ethan J. Kubatko and Joannes J. Westerink and Clint Dawson},
journal = {Journal of Computational Physics},
pages = {832-848},
title = {Semi discrete discontinuous {G}alerkin methods and stage-exceeding-order, strong-stability-preserving {R}unge-{K}utta time discretizations},
volume = {222},
year = {2007}
}
@article{lenferink1989,
author = {H W. J. Lenferink},
journal = {Numerische Mathematik},
pages = {213-223},
title = {Contractivity-preserving explicit linear multistep methods},
volume = {55},
year = {1989}
}
@article{lenferink1991,
author = {H W. J. Lenferink},
journal = {Math. Comp.},
pages = {177-199},
title = {Contractivity-preserving implicit linear multistep methods},
volume = {56},
year = {1991}
}
@article{liu2008,
author = {Y. Liu and C.-W. Shu and M. Zhang},
journal = {Journal of Computational Mathematics},
note = {to appear},
title = {Strong stability preserving property of the deferred correction time discretization}
}
@unpublished{macdonald2007,
author = {Colin MacDonald and Sigal Gottlieb and Steven Ruuth},
title = {A numerical study of diagonally split {R}unge-{K}utta methods for {PDE}s with discontinuities}
}
@mastersthesis{macdonaldmsthesis,
author = {Colin B. Macdonald},
month = {August},
school = {Simon Fraser University},
title = {Constructing High-Order {R}unge-{K}utta Methods with Embedded Strong-Stability-Preserving Pairs},
year = {2003},
}
@article{ruuth2004,
author = {S. J. Ruuth and R. J. Spiteri},
journal = {SIAM Journal of Numerical Analysis},
pages = {974-996},
title = {High-order strong-stability-preserving {R}unge-{K}utta methods with downwind-biased spatial discretizations},
volume = {42},
year = {2004}
}
@article{ruuth2006,
author = {Steven Ruuth},
journal = {Math. Comp.},
pages = {183-207},
title = {Global optimization of explicit strong-stability-preserving {R}unge-{K}utta Methods},
volume = {75},
year = {2006}
}
@article{ruuth2005,
author = {Steven J. Ruuth and Willem Hundsdorfer},
journal = {Journal of Computational Physics},
pages = {226-248},
title = {High-order linear multistep methods with general monotonicity and boundedness properties},
volume = {209},
year = {2005}
}
@article{ruuth2001,
author = {Steven J. Ruuth and Raymond J. Spiteri},
journal = {Journal of Scientific Computation},
pages = {211-220},
title = {Two Barriers on Strong-Stability-Preserving Time Discretization Methods},
volume = {17},
year = {2002}
}
@article{shu1988,
author = {C.-W. Shu and S. Osher},
journal = {Journal of Computational Physics},
pages = {439-471},
title = {Efficient implementation of essentially non-oscillatory shock-capturing schemes},
volume = {77},
year = {1988},
}
@incollection{shu2002,
author = {Chi-Wang Shu},
booktitle = {Collected Lectures on the Preservation of Stability under discretization},
publisher = {SIAM: Philadelphia, PA},
title = {A survey of strong stability-preserving high-order time discretization methods},
year = {2002}
}
@article{shu1988b,
author = {Chi-Wang Shu},
journal = {SIAM J. Sci. Stat. Comp.},
pages = {1073-1084},
title = {Total-variation diminishing time discretizations},
volume = {9},
year = {1988},
}
@article{spijker1983,
author = {M. N. Spijker},
journal = {Numerische Mathematik},
pages = {271-290},
title = {Contractivity in the numerical solution of initial value problems},
volume = {42},
year = {1983},
}
@article{spijker1984,
author = {M. N. Spijker},
journal = {BIT},
pages = {656-666},
title = {On the relation between stability and contractivity},
volume = {24},
year = {1984},
}
@article{spijker2007,
abstract = {For Runge-Kutta methods and linear multistep methods, much attention has been paid, in the literature, to special nonlinear stability properties indicated by the terms total-variation-diminishing (TVD), strong-stability-preserving (SSP), and monotonicity. Stepsize conditions, guaranteeing these properties, were studied, e. g., by Shu and Osher [J. Comput. Phys., 77 (1988), pp. 439-471], Gottlieb, Shu, and Tadmor [SIAM Rev., 43 (2001), pp. 89-112], Hundsdorfer and Ruuth [Monotonicity for Time Discretizations, Dundee Conference Report NA/217 2003, University of Dundee, Dundee, UK, 2003, pp. 85-94], Higueras [J. Sci. Comput., 21 (2004), pp. 193-223] and [SIAM J. Numer. Anal., 43 (2005), pp. 924-948], Spiteri and Ruuth [ SIAM J. Numer. Anal., 40 (2002), pp. 469-491], Gottlieb [ J. Sci. Comput., 25 (2005), pp. 105-128], and Ferracina and Spijker [ SIAM J. Numer. Anal., 42 (2004), pp. 1073-1093] and [ Math. Comp., 74 (2005), pp. 201-219]. In the present paper, we obtain a special stepsize condition guaranteeing the above properties, for a generic numerical process. This condition is best possible in a well defined and natural sense. It is applicable to the important class of general linear methods, and it can also be used to answer some open questions, for methods of which the above stability properties were studied earlier.},
author = {Spijker, M. N.},
doi = {DOI 10.1137/060661739},
journal = {Siam Journal On Numerical Analysis},
keywords = {initial value problem; method of lines; ordinary differential equations; general linear method; total variation diminishing; strong stability preservation; monotonicity},
pages = {1226-1245},
title = {Stepsize conditions for general monotonicity in numerical initial value problems},
volume = {45},
year = {2007},
}
@article{spijker1985,
author = {M. N. Spijker},
journal = {Mathematics of Computation},
pages = {377-392},
title = {Stepsize Restrictions for stability of one-step methods in the numerical solution of initial value problems},
volume = {45},
year = {1985}
}
@article{spiteri2002,
author = {Raymond J. Spiteri and Steven J. Ruuth},
journal = {SIAM Journal of Numerical Analysis},
pages = {469-491},
title = {A New Class of Optimal High-Order Strong-Stability-Preserving Time Discretization Methods},
volume = {40},
year = {2002}
}
@article{spiteri2003,
author = {Raymond J. Spiteri and Steven J. Ruuth},
journal = {Mathematics and Computers in Simulation},
pages = {125-135},
title = {Nonlinear Evolution Using Optimal Fourth-Order Strong-Stability-Preserving {R}unge-{K}utta Methods},
volume = {62},
year = {2003}
}
@article{wang2007,
author = {Rong Wang and Raymond J. Spiteri},
doi = {10.1137/050637868},
journal = {SIAM Journal on Numerical Analysis},
keywords = {stability analysis; Runge-Kutta methods; WENO method; strong stability preservation},
number = {5},
pages = {1871-1901},
publisher = {SIAM},
title = {Linear Instability of the Fifth-Order WENO Method},
url = {http://link.aip.org/link/?SNA/45/1871/1},
volume = {45},
year = {2007},
}