Implicit Second-derivative Runge-kutta Collocation Methods of Uniformly Accurate Order 3 and 4 for the Solution of Systems of Initial Value Problems

We consider the construction of a new class of implicit Second-derivative Runge-Kutta collocation methods based on intra-step nodal points of Chebyshev-Gauss-Lobatto type, designed for the numerical solution of systems of initial value equations and show how they have been implemented in an efficient parallel computing environment. We also discuss the difficulty associated with large systems and how, in this case, one must take advantage of the second derivative terms in the methods. We involve the introduction of collocation at the two end points of the integration interval in addition to the Gaussian interior collocation points and also the introduction of a different class of basic second derivative methods. With these modifications, fewer function evaluations per step are achieved. The stability properties of these methods are investigated and numerical results are given for each method.