Parallel Second Order Runge – Kutta new method of the Geometric Mean

Runge - Kutta methods are one of the best methods for numerically solving (ODEs), and the search for better methods is always up to date [1]. Our concern here is with present a new method for solving Initial Value Problems (IVPs) using a mixing between techniques and formulas and obtain a new formula suitable for parallel computers, (see [2]). As we know a first step toward developing a parallel algorithm for the numerical solution of Initial Value Problems (IVPs), how we might widen the front of computation .The predictor – corrector (PC) methods of numerical integration provide a means for doing this, (see [3,4]) ."Evans Introduce a new Runge - Kutta method using the Geometric mean (GM) formula [5]" (see [6]). Here we collected these ideas and using the Implicit Runge - Kutta methods (IRK) which can be derived directly from Explicit Runge - Kutta methods (ERK) , these implicit methods “ that were derived “ , it is "the (backward) form of the explicit (forward) form [7]", to present our new parallel method which we called (PPCGM2) formula.