RungeKutta - Runge-Kutta Method The Runge-Kutta method...

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Runge-Kutta Method The Runge-Kutta method, discovered by German mathematicians Carl Runge and Martin Kutta, is a powerful fourth order method (i.e. the error is at most a fixed constant times the fourth power of the step size, h ), thus a great improvement over the Improved Euler’s method. To justify this scheme, we need to review Simpson’s Rule for estimating a definite integral. Simpson’s Rule Suppose we have a definite integral of the form Z a + h a f ( x ) dx. Two ways of approximating this area are (1) to use a trapezoid and (2) to use a rectangle whose height is determined by the midpoint a + h/ 2 . If we call these approximations T and M , respectively, then their formulas are: T = h 2 ( f ( a ) + f ( a + h )) and M = hf ( a + h/ 2) . It can be shown that the error in the trapezoid rule is given by E T = - 1 24 h 3 f 00 ( a ) + O ( h 4 ) and the error in the midpoint rule is given by E M = 1 12 h 3 f 00 ( a ) + O ( h 4 ) where O ( h 4 ) means the remaining error terms are proportional to at least h 4 . Simpson’s rule is precisely the right weighted average of T
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