ABSTRACT
The construction of multistep formulae from Taylor expansions is somewhat easier than the derivation of Runge–Kutta (RK) processes. Inevitably, this method is devoted to the pursuit of algebraic order rather than any other attribute such as absolute stability. A particularly convenient method of generating related formulae of different orders is based on numerical quadrature. The families of formulae so developed share certain useful characteristics which makes their computer implementation rather attractive. Quadrature formulae, which cover an interval beyond the extent of the data, are known as open formulae. The use of quadrature formulae for multistep processes can be extended beyond the Adams families which can be considered as a special case of formulae based on the integral relation. The chapter considers the systematic derivation and also the implementation of the Adams processes. These are even more ancient than RK processes, dating back to the middle of the nineteenth century.
