Multivalue second derivative collocation methods

We introduce multivalue second derivative collocation methods for the numerical solution of stiff ordinary differential equations, also arising from the spatial discretization of time dependent partial differential equations. The uniform order of convergence of the methods is discussed and continuous order conditions are derived. We construct methods of orders up to eight with desirable stability properties. Numerical experiments are given, validating the theoretical results and illustrating the efficiency and capability of the proposed methods in solving stiff problems without any reduction of the order of convergence, unlike A–stable Runge–Kutta methods.