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.


  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

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Information & Authors


Published In

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


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


  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



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

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