(A) (i): When numerically computing the solution to an ODE
that involves higher-than first-order derivatives, the higher-order
terms must be reduced to 1st-order terms. This can be accomplished
by introducing new substituted variables such as
(ii) Thus the example ODE,
(B) (i) The incrementing rule for the Euler method of numerical
integration is:
(ii) If the stepsize h is too large, this piecewise-linear approximation to the solution will be inaccurate where the true solution has significant second derivatives (which this method ignores). The solution may ``blow up."
(iii) If the stepsize h is made too small, the computation may take too long and the round-off error (truncation of floating-point numbers depending on the word length of the machine) will accumulate too much, making the solution inaccurate or even unstable.
(iv) The primary advantage of the fourth-order Runge-Kutta method over the Euler method for numerical integration of ODEs is that its accumulated error depends on the stepsize h as its 4th power, h4, whereas in the Euler method the accumulated error is simply linear in h. Thus, for example, reducing the stepsize h by half would only reduce the accumulated error by half in the Euler method, but it would reduce it 16-fold in the Runge-Kutta method.