Convergence and stability of variable-stepsize variable-formula multistep multiderivative methods

During the numerical integration of a system of first order differential equations, practical algorithms which use linear multistep formulas try to keep the estimated local truncation error smaller than a user-supplied tolerance. This is usually achieved by allowing for changes in the stepsize and/or the formula being used. As a result, the algorithm becomes a variable-stepsize variable-formula method (VSVFM);A general definition of a VSVFM is given which places no restrictions on how the stepsizes can be changed when using the method, but instead, it limits the multistep formulas which can be included in the method. This definition also allows for the use of higher derivative multistep formulas. A slightly more stringent definition of stability than was used by Gear and Tu in 1974 is given;Theorems are proved which give necessary and sufficient conditions for both stability and convergence of a VSVFM. An extension to a theorem given by Crouziex and Lisbona in 1984 is derived which gives sufficient conditions for a VSVFM to be stable. Convergence and stability are shown for several VSVFMs