In this thesis, we study the contractivity preserving, high order, time discretization methods for solving non-stiff ordinary differential equations. We construct a class of one-step, explicit, contractivity preserving, multi-stage, multi-derivative, Hermite-Birkhoff-Taylor methods of order p=5,6, ..., 15, that we denote by CPHBT, with nonnegative coefficients by casting s-stage Runge-Kutta methods of order 4 and 5 with Taylor methods of order p-3 and p-4, respectively. The constructed CPHBT methods are implemented using an efficient variable step algorithm and are compared to other well-known methods on a variety of initial value problems. The results show that CPHBT methods have larger regions of absolute stability, require less function evaluations and hence they require less CPU time to achieve the same accuracy requirements as other methods in the literature. Also, we show that the contractivity preserving property of CPHBT is very efficient in suppressing the effect of the propagation of discretization errors when a long-term integration of a standard N-body problem is considered. The formulae of 49 CPHBT methods of various orders are provided in Butcher form.

In this thesis, we construct a new optimal contractivity-preserving (CP) explicit, 2-step, 6-stage, 6-derivative, Hermite--Birkhoff--Obrechkoff method of order 13, denoted by HBO(13) with nonnegative coefficients, for solving nonstiff first-order initial value problems y'=f(t,y), y(t_0)=y_0. This new method is the combination of a CP 2-step, 6-derivative, Hermite--Obrechkoff of order 9, denoted by HO(9), and a 6-stage Runge-Kutta method of order 5, denoted by RK(6,5). The new HBO(13) method has order 13. We compare this new method, programmed in Matlab, to Adams-Bashforth-Moulton method of order 13 in PECE mode, denoted by ABM(13), by testing them on several frequently used test problems, and show that HBO(13) is more efficient with respect to the CPU time, the global error at the endpoint of integration and the relative energy error. We show that the new HBO(13) method has a larger scaled interval of absolute stability than ABM(13) in PECE mode.