Abstract

Strong stability preserving (SSP) methods are designed primarily for time integration of nonlinear hyperbolic PDEs, for which the permissible SSP step size varies from one step to the next. We develop the first SSP linear multistep methods (of order two and three) with variable step size, and prove their optimality, stability, and convergence. The choice of step size for multistep SSP methods is an interesting problem because the allowable step size depends on the SSP coefficient, which in turn depends on the chosen step sizes. The description of the methods includes an optimal step-size strategy. We prove sharp upper bounds on the allowable step size for explicit SSP linear multistep methods and show the existence of methods with arbitrarily high order of accuracy. The effectiveness of the methods is demonstrated through numerical examples.

Keywords

  1. strong stability preservation
  2. monotonicity
  3. linear multistep methods
  4. variable step size
  5. time integration

MSC codes

  1. Primary
  2. 65M20; Secondary
  3. 65L06

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
S. Gottlieb, D. I. Ketcheson, and C.-W. Shu, Strong Stability Preserving Runge--Kutta And Multistep Time Discretizations, World Scientific, Haekensack, NJ, 2011.
2.
E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd ed., Springer Ser. Comput. Math., Springer, Berlin, 1993.
3.
W. Hundsdorfer and S. J. Ruuth, On monotonicity and boundedness properties of linear multistep methods, Math. Comp., 75 (2005), pp. 655--672.
4.
W. Hundsdorfer, S. J. Ruuth, and R. J. Spiteri, Monotonicity-preserving linear multistep methods, SIAM J. Numer. Anal., 41 (2003), pp. 605--623.
5.
David I. Ketcheson, Computation of optimal monotonicity preserving general linear methods, Math. Comp., 78 (2009), pp. 1497--1513.
6.
D. I. Ketcheson, K. Mandli, A. J. Ahmadia, A. Alghamdi, M. Quezada de Luna, M. Parsani, M. G. Knepley, and M. Emmett, PyClaw: Accessible, extensible, scalable tools for wave propagation problems, SIAM J. Sci. Comput., 34 (2012), pp. C210--C231.
7.
D. I. Ketcheson, M. Parsani, and R. J. LeVeque, High-order wave propagation algorithms for hyperbolic systems, SIAM J. Sci. Comput., 35 (2013), pp. A351--A377.
8.
M. R. S. Kulenović and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, FL, 2002.
9.
H. W. J. Lenferink, Contractivity-preserving explicit linear multistep methods, Numer. Math., 55 (1989), pp. 213--223.
10.
H. W. J. Lenferink, Contractivity-preserving implicit linear multistep methods, Math. Comp., 56 (1991), pp. 177--199.
11.
A. Németh and D. I. Ketcheson, Existence and Optimality of Strong Stability Preserving Linear Multistep Methods: A Duality-Based Approach, preprint, arXiv:1504.03930, 2015.
12.
S. J. Ruuth and W. Hundsdorfer, High-order linear multistep methods with general monotonicity and boundedness properties, J. Comput. Phys., 209 (2005), pp. 226--248.
13.
A. Schrijver, Theory of Linear and Integer Programming, Wiley Chichester, England, 1998.
14.
B. van Leer, Towards the ultimate conservative difference scheme. IV. A new approach to numerical convection, J. Comput. Phys., 23 (1977), pp. 276--299.
15.
P. Woodward and P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54 (1984), pp. 115--173.
16.
X. Zhang and C.-W. Shu, On maximum-principle-satisfying high order schemes for scalar conservation laws, J. Comput. Phys., 229 (2010), pp. 3091--3120.

Information & Authors

Information

Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 2799 - 2832
ISSN (online): 1095-7170

History

Submitted: 16 April 2015
Accepted: 21 June 2016
Published online: 8 September 2016

Keywords

  1. strong stability preservation
  2. monotonicity
  3. linear multistep methods
  4. variable step size
  5. time integration

MSC codes

  1. Primary
  2. 65M20; Secondary
  3. 65L06

Authors

Affiliations

Funding Information

TAMOP : 4.2.2.A-11/1/KONV-2012-0012

Funding Information

King Abdullah University of Science and Technology http://dx.doi.org/10.13039/501100004052 : 1

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.