EXPLANATION FILE OF PROGRAM TEQUDIF1 ==================================== The Runge-Kutta Method ---------------------- We present here the Runge-Kutta method of order 4 to integrate an ODE of order 1: Y' = F(X, Y) The development of Y around x coincidates with its Taylor development n of order 4: y = y + h y' + (h^2/2) y" + (h^3/6) y"' + (h^4/24) y"" n+1 n n n n n Other orders may also be used for the Runge-Kutta method, for instance the Euler's method is a Runge-Kutta of order one. Order 4 is often used because it is a good compromise between speed and accuracy. From a current point (x , y ), the next point is determined by: n n K = h f(x , y ) 0 n n k = h f(x + h/2, y + K /2) 1 n n 0 k = h f(x + h/2, y + K /2) 2 n n 1 k = h f(x + h, y + K ) 3 n n 2 and y = y + (1/6)(K + 2 K + K ) n+1 n 0 1 3 It can be proved that the error at each step is about h^5. The only drawback is the time of calculation: at each step, four function evaluations are necessary. From [BIBLI 03]. ------------------------------------------------------------- End of file rkutta.txt.