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2.2.2 Runge-Kutta Methods     continued...

If we increase n and tabulate the solution at , we obtain the results in Table 5, and we see that the error, as h approaches zero.

Table 5: Approximate solution using the Euler-Cauchy method.

A basic assumption in the derivation of the family of Runge-Kutta formulas (34) was that the solution had three continuous derivatives. What if a formula of order 2 is used to solve an initial value problem whose solution has only two continuous derivatives, but not three. Examination of the local truncation error shows that the formula is then of order 1 and convergence is and not . The point here is that higher order procedures can be used on problems whose solutions are not sufficiently smooth, but their rate of convergence may be reduced.

A common example of problems whose solutions will not be smooth are those where the coefficients have a jump discontinuity at some point in the range of integration. In solving such problems numerically, integration should not be performed across the discontinuity. For example, suppose we are solving the problem

A good procedure would be to integrate from t=0 to t=1 and then from t=1 to t=2. On each subinterval, the differential equation has smooth solutions and convergence rates will be as advertised.