Efficient sixth order methods for nonlinear oscillation problems
By applying two-step (hybrid) methods to solve initial value problems of the kind (1) \(\ddot x=f(t,x)\), \(x(0)\), \(\dot x(0)\) given with nonlinear f, one has to solve a nonlinear algebraic system at each step. This may be done by a modified Newton iteration scheme. The author derives a class of fourth order, P-stable methods which require only two function evaluation per iteration and one matrix factorization at each step. For a definition of P-stability see \textit{J. D. Lambert} and \textit{I. A. Watson} [J. Inst. Math. Appl. 18, 189-202 (1976; Zbl 0359.65060)]. Furthermore there is propsed an extension to sixth order P-stable methods which require only three function evaluation per iteration. These methods have been implemented on a new variable step, local error controlling code.
variable step, local error controlling code
scientific article; zbMATH DE number 4092688