Question

Q. What is difference between single step and multi step method?

Answer 1

Answer on Question #76735 – Math – Quantitative Methods

Question

What is difference between single step and multi step method?

Solution

To approximate the solution to the 1st order IVP:

$$y' = f(x, y), \qquad y(x_0) = y_0$$

we seek the approximate value of the function $y(x)$ at discrete points $x_i$:

$$y_i \approx y(x_i), \qquad x_i = x_0 + i \cdot h, \qquad h - \text{step size}$$

The accuracy can be numerically estimated: by halving step size (multi step method) or by increasing order (single step method).

The one-step method is that at each step of the method the accuracy of the function is determined by the difference in values between the results of the p-th and higher-order methods. This method estimates the error of the lower order scheme.

The multi-step method is based on estimating the difference between the approximate value of the function obtained using step size $h$ and the approximate value of the function obtained using step size $h/2$. Thus, it is required to calculate three times the approximate value of the function (once with step size $h$ and twice with step size $h/2$). This method requires much more calculations.

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