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.

Certain numerical methods for initial value problems have as stability function the nth partial sum of the exponential function. We study the stability region, that is, the set in the complex plane over which the nth partial sum has at most unit modulus. It is known that the asymptotic shape of the part of the stability region in the left half-plane is a semidisc. We quantify this by providing discs that enclose or are enclosed by the stability region or its left half-plane part. The radius of the smallest disc centred at the origin that contains the stability region (or its portion in the left half-plane) is determined for 1 ≤ n ≤ 20. Bounds on such radii are proved for n ≥ 2; these bounds are shown to be optimal in the limit n → +∞. We prove that the stability region and its complement, restricted to the imaginary axis, consist of alternating intervals of length tending to π, as n → ∞. Finally, we prove that a semidisc in the left half-plane centred at the origin and with vertical boundary lying on the imaginary axis is included in the stability region if and only if n ≡ 0 mod 4 or n ≡ 3 mod 4. The maximal radii of such semidiscs are exactly determined for 1 ≤ n ≤ 20.