Classic Runge-Kutta method

The classic Runge-Kutta method (according to Carl Runge and Martin Wilhelm Kutta ) is a special, explicit 4-stage Runge-Kutta method for the numerical solution of initial value problems ( ordinary differential equations ). An abbreviation of this procedure is RK4 . Runge was the first (1895) to specify a multi-stage process and Kutta the general form of explicit s-stage processes.

The classic Runge-Kutta method uses - like most of the numerical solution methods for differential equations - the approach of approximating derivatives ( differential quotients ) by difference quotients. The errors that necessarily occur with non-linear functions (all higher terms of the Taylor expansion are neglected) can be reduced by suitable combinations of different difference quotients. The classic Runge-Kutta method is such a combination that compensates for discretization errors up to the third derivative.

Details

The classic Runge-Kutta method averages four auxiliary slopes (red) in each step

Be

{\ displaystyle y '(t) = f \ left (t, y (t) \ right), \ quad y (t_ {0}) = y_ {0}, \ quad y \ colon \ mathbb {R} \ to \ mathbb {R} ^ {d}}

a first order initial problem.

With the step size , the classic Runge-Kutta method for calculating the approximation has the procedural function ${\ displaystyle h}$ ${\ displaystyle u_ {i + 1} \ approx y (t_ {i + 1})}$

{\ displaystyle \ Phi (t_ {i}, u_ {i}, h, f) = {\ frac {1} {6}} k_ {1} + {\ frac {1} {3}} k_ {2} + {\ frac {1} {3}} k_ {3} + {\ frac {1} {6}} k_ {4}}

With

{\ displaystyle {\ begin {aligned} k_ {1} & = f (t_ {i}, u_ {i}), \\ k_ {2} & = f (t_ {i} + {\ frac {h} { 2}}, u_ {i} + {\ frac {h} {2}} k_ {1}), \\ k_ {3} & = f (t_ {i} + {\ frac {h} {2}} , u_ {i} + {\ frac {h} {2}} k_ {2}), \\ k_ {4} & = f (t_ {i} + h, u_ {i} + hk_ {3}). \ end {aligned}}}

The recursion equation for calculating the approximation is then

{\ displaystyle u_ {i + 1} = u_ {i} + h \ cdot \ Phi (t_ {i}, u_ {i}, h, f) = u_ {i} + h \ cdot {\ frac {1} {6}} \ left (k_ {1} + 2k_ {2} + 2k_ {3} + k_ {4} \ right), \ quad i = 0.1, \ dots}

The procedure requires four evaluations of the function in each step of the recursion . For something that is continuously differentiable at least four times , a Taylor expansion according to the step size shows that the classic Runge-Kutta method is a method with a consistency order of 4. ${\ displaystyle f}$ ${\ displaystyle f}$ ${\ displaystyle h}$

The characteristic coefficients of the method can be summarized in a Butcher tableau as follows:

${\ displaystyle {\ begin {array} {c | cccc} 0 &&&& \\ 1/2 & 1/2 &&& \\ 1/2 & 0 & 1/2 && \\ 1 & 0 & 0 & 1 & \\\ hline & 1/6 & 1/3 & 1/3 & 1/6 \ end {array} }}$

literature

Ernst Hairer, Nørsett, Syvert P., Gerhard Wanner: Solving Ordinary Differential Equations . Volume 1: Nonstiff Problems . 2nd revised edition. Springer Verlag, Berlin et al. 1993, ISBN 3-540-56670-8 ( Springer series in computational mathematics 8), (also reprint: ibid 2008, ISBN 978-3-642-05163-0 ).
Peter Deuflhard , Folkmar Bornemann : Numerical Mathematics . Volume 2: Ordinary Differential Equations . 2nd completely revised and enlarged edition. de Gruyter, Berlin 2002, ISBN 3-11-017181-3 .