  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.
