Topic 2: The pendulum

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PHY321F — cp 2005 31 D.2 Runge-Kutta methods The simple methods we have considered are only correct to first order in h. Obviously, higher order methods can be expected to offer benefits (although this might not extend to accuracy). One way of generating such methods leads to a class of Runge-Kutta methods. By way of example, suppose we solve a system by the Euler method to get dy(x) dx = f(x, y) y(x + h) = y(x) + h f(x, y) The derivative over the interval of size h is taken from the initial point. A better value, perhaps, would be to take the value at the centre of the interval. The value of y at the centre of the interval can be estimated from the above Euler step This then gives a solution k 1 = h f(x, y(x)) k 2 = h f(x + h 2 , y(x) + 1 2 k 1) y(x + h) = y(x) + k 2 This method turns out to be second order, so that the error term is O(h 3 ) This process can be continued: the classic Runge-Kutta scheme is the fourth order formula k 1 = h f(x, y(x)) k 2 = h f(x + h 2 , y(x) + 1 2 k 1) k 3 = h f(x + h 2 , y(x) + 1 2 k 2) k 4 = h f(x + h 2 , y(x) + k 3) y(x + h) = y(x) + k 1 6 + k 2 3 + k 3 3 + k 4 6 This is easily programmed, and thus is often used as a workhorse fixedstepsize integration formula.
PHY321F — cp 2005 32 More control over the process is achieved if the error can be monitored. Suppose we have two Runge-Kutta formulae, with solutions y(x + h) of order h n+1 and ŷ(x + h) of order h n , Then the difference y − ŷ gives an estimate of the error in y. This is especially useful if we can obtain both y and ŷ from the same set of Runge-Kutta steps. (Such formulae are known as embedded Runge-Kutta formulae; the additional work does not take additional expensive evaluations of the function f). Once we have an estimate of the error, it can be used to control the stepsize so that some maximum error per step is not exceeded. This leads to a set of adaptive Runge-Kutta methods. A modern adaptive RK code is rksuite (or rksuite90), available from netlib. D.3 Other methods D.3.1 Predictor-corrector methods The Euler and Runge-Kutta methods extrapolate from one point to the next. Higher accuracy can be obtained by using several previous steps. These multistep methods are usually implemented as predictor-corrector schemes. An extrapolation is made to the next point using an explicit multistep formula. The value obtained is then used in an implicit multistep formula to correct this prediction. This scheme also permits error control by adaptive step sizing. A standard PC code is vode (and related codes), available from netlib. These methods give high accuracy integrators, but are more complicated and fussy to program than RK. D.3.2 Extrapolation methods Burlisch-Stoer methods (see, e.g. Numerical Recipes) use Richardson extrapolation to the limit h → 0 in order to improve accuracy. Press. et. al seem to think that these methods are about to replace PC methods (if they haven’t already). Others disagree.
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Topic 2: The pendulum

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