Page 96 - Standard Handbook Of Petroleum & Natural Gas Engineering

Numerical Methods 85

therefore

1
AY = 2 +By,)

and

YI = YO + 'Y
Another improvement on the basic Euler method is to approximate the slope
in the middle of the interval by the average of the slopes at the end points, or

This form is a closed-type formula since it does not allow direct steps from xi to
xirl, but uses the basic Euler's method to estimate yitl, thus

The Runge-Kutta method takes the weighted average of the slope at the left
end point of the interval and at some intermediate point. This method can be
extended to a fourth-order procedure with error 0 AX)^ and is given by

where Ayo = f(xi, y,)Ax

Runge-Kutta formulas of the sixth and eighth orders are also available, but less
commonly used.
If two values of yi+l are calculated, yi+l by using one step between xi and xi+l
with Ax, and y,,, by taking two steps with Ax/2, the estimate of the truncation
error is