A new efficient algorithm for solving differential-algebraic systems using implicit backward differentiation formulas

The backward differentiation formulas (BDF), of order 1 up to 6 are described as they are applied to a system of differential algebraic equations. The BDF method is compared to the Gear-Nordsieck method, and is shown to be more efficient, more flexible in the selection of variables for prediction and error control, and more stable under conditions of rapidly varying Δt. For Δt fixed, the two methods are equivalent but for Δt varying they are not equivalent. Numerical experiments are described which demonstrate that the Gear-Nordsieck and BDF methods are unstable under rapidly changing Δt, but BDF is more stable. The two methods are distinguished numerically by identifying the modification of the Gear-Nordsieck method which makes it equivalent to the BDF method even if Δt changes. The computational advantage of using backward differences Δx, instead of the Nordsieck vector, for storing the backward-time information is treated by giving an operations count which shows the BDF using backward Δx's is more efficient. Finally, additional numerical evidence is given to support the use of variable order methods and the use of higher order methods.