The control of order and steplength for backward differentiation methods

Backward differentiation methods are used extensively for integration of stiff systems of ordinary differential equations, but most implementations are inefficient when some of the eigenvalues of the Jacobi matrix are close to the imaginary axis. For these problems the performance of backward differentiation methods can be improved considerably by application of the instability test and reaction which is described in this paper. During instability the local truncation error oscillates rapidly with increasing magnitude. This property is used in the instability test. When instability is detected the order is lowered as much as possible without reducing the steplength.The instability test and reaction is derived from a simplified analysis of integration of linear systems of differential equations, and the performance is verified for a number of linear test problems.This work was supported by »Statens Teknisk-Videnskabelige Forskningsråd« under grant no. 516-6537. E-368.