1 Week 11 : 4 and 6 November 2008

We now discuss the theory of fixed–order constant–step–size linear multistep methods. This classical theory is presented in [HNW93] Sect. III.1–4 (nons-tiff) and [HW96] Sect. V.1 (stiff), as well as [Gea71] Chap. 7, [Lam91] Chap. 3, [Sha94] Chap. 4, and many other sources. While multistep methods are rarely used in practice with fixed order and constant step size, this theory is useful because its conclusions can be derived without numerical calculations and because it accurately predicts the behavior of the variable-step variable–order (VSVO) formulas used in codes like cvode in two common situations. First, when the step sizes vary arbitrarily within an upper bound h n /h n−1 = ω n ≤ Ω on the step ratios ω n. For example, one can prove that 2–step BDF with variable step sizes is stable as long as ω n < Ω = 1 + √ 2, while 5–step BDF has a more stringent requirement Ω = 1.003. (Here " stability " means that the numerical solution is a Lipschitz function of perturbations to the right–hand side f and starting values u 0 , u 1 ,. .. , u k−1 in the precise sense defined above.) Second, when the step sizes increase faster than Ω but their variation is smooth rather than completely arbitrary. 2 Overview The classical theory analyzes multistep methods u n+1 + α 0 u n + · · · + α k−1 u n−k+1 = h(β −1 f n+1 + β 0 f n + · · · + β k−1 f n−k+1) (1) with constant step size h and order k, so the coefficients α and β are fixed dimensionless numbers. The stability, consistency and (linear) stiff stability properties of these methods can be read off from two polynomials ρ(z) = α −1 z k + α 0 z k−1 + · · · + α k−1 and σ(z) = β −1 z k + β 0 z k−1 + · · · + β k−1 (where we usually normalize by putting α −1 = 1). Nonlinear stiff stability properties are mostly irrelevant because of the Dahlquist barrier theorems: Theorem 1 A stable k–step method has order p ≤ k+2 if k is even, p ≤ k+1 if k is odd, and p ≤ k if it is explicit.