Stability and Convergence of Variable Order Multistep Methods

A method based on variable step Adams formulas is shown to be stable for any order changing scheme. A method based on the Nordsieck form of Adams formulas, however, is shown to be stable only if the step size and order are fixed for p steps following a change to an r-step formula, where p is r or r + 1 depending on the algorithm used to interpolate the necessary higher derivatives. Finally, general methods based on consistent and strongly stable multistep and multivalue formulas are shown to be stable if the formula is fixed for a certain number of steps following each formula change and if step size changes are small. This number is independent of the differential equation and the step sizes.