A new method using the Forward Backward technique with Contra Harmonic mean formula

We introduce a new method for solving Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs), by making a mixing between the Forward (Predictor) – Backward (Corrector) technique and used it in the Contra Harmonic mean formula, this new method give us a parallelism in numerical calculations and it is more accurate than the old classical Runge – Kutta formula of the same order.

Here, we present new method for solving (IVPs) combining between previously techniques and obtain a new formula suitable for parallel computations. In 1967 Miranker [1], present the concern of computation front "the imaginary straight line that separate the values are next to be computed (by some numerical algorithms) from all previously computing value problems", which is not give full advantage of multi processers computers, because it too narrow. By predictorcorrector (PC) method we made an expansion to the computation front.
In 1993 Evans [2] presented the Contra Harmonic mean formula. In 2020 Jasim,M.D [3,4] introduced new methods by using more than one mean with the (PC) method. Our new method namely ( PPCCoM1).

2.The Contra Harmonic mean formula.
The Contra Harmonic mean formula has the form, Where, we regarded ߔ a function only of w.

3.New method namely PPCCoM1 :
The new method PPCCoM1 calculates ‫ݓ‬ ାଵ depending on ‫ݓ‬ ିଵ and the computation of ‫ݓ‬ depending on ‫ݓ‬ ିଵ and ‫ݓ‬ , which has form, And, Illustrated the computation process of PPCCoM1 mode in figure 1 below, Figure 1: information of PPCCoM1-mode.

4.Derivation of PPCCoM1-method.
Predictor Contra Harmonic mean method of two stage second order has form, To derive our Forward (predictor) PPCCoM1, expansion of ‫ݐ‬ ଵ and ‫ݐ‬ ଶ gives Where, ‫(‬ℎ) is the local truncation error.
We get, To get the corrector part, from the backward formula, "replace h by −h " (see [6]).
Expansion of ‫ݒ‬ ଵ and ‫ݒ‬ ଶ in (16) gives, Substitute (17) in (15) we get , We have the same problem in equation (11), so equation (18) written as, Where,  [5] given as, Written the difference between equation (19) and (20)  Which is two equations with five parameters, so we get three freedom degree.
Where, ‫ݓ‬ ାଵ represent the forward form, ‫ݓ‬ is the backward form.

5.The stability region of PPCCoM1
AS it know, the Runge -Kutta methods are stable, and it have a good quite of stability, when we choose a small step size.
Examine the stability of the forward part of PPCCoM1, To find the interval of absolute stability, we used the test equation ẃ = ‫ݓߣ‬ [9,10], Substitute equations (24) and (25) in (13) we get , The last equation (27) satisfies the absolute stability condition if |߬ ଵ | < 1 where ࣴ identify the condition when ࣴ ∊ ( -0.6478, 0 ) which represents the stability region of this method. As we see in table1, Comparison of the exact solution (in column 2) with our PPCCoM1 (in column 4) shows that, the PPCCoM1 is stable in the interval of integration and has a good accuracy in results, comparing with the exact solution. The conclusion is clearly observed in the last column 6 (the absolute difference between the exact solution and the backward solution of PPCCoM1) in table1.