Mathematical Methods for Physics and Engineering - Matematica.NET

NUMERICAL METHODSSteps (ii) and (iii) can be iterated to improve further the approximation to theaverage value of dy/dx, but this will not compensate for the omission of higherorderderivatives in the forward difference formula.Many more complex schemes of prediction and correction, in most casescombining the two in the same process, have been devised, but the reader isreferred to more specialist texts for discussions of them. However, because itoffers some clear advantages, one group of methods will be set out explicitly inthe next subsection. This is the general class of schemes known as Runge–Kuttamethods.27.6.4 Runge–Kutta methodsThe Runge–Kutta method of integratingdy= f(x, y) (27.74)dxis a step-by-step process of obtaining an approximation for y i+1 by starting fromthe value of y i . Among its advantages are that no functions other than f are used,no subsidiary differentiation is needed and no additional starting values need becalculated.To be set against these advantages is the fact that f is evaluated using somewhatcomplicated arguments and that this has to be done several times for each increasein the value of i. However, once a procedure has been established, for exampleon a computer, the method usually gives good results.The basis of the method is to simulate the (accurate) Taylor series for y(x i + h),not by calculating all the higher derivatives of y at the point x i but by takinga particular combination of the values of the first derivative of y evaluated ata number of carefully chosen points. Equation (27.74) is used to evaluate thesederivatives. The accuracy can be made to be up to whatever power of h is desired,but, naturally, the greater the accuracy, the more complex the calculation, and,in any case, rounding errors cannot ultimately be avoided.The setting up of the calculational scheme may be illustrated by consideringthe particular case in which second-order accuracy in h is required. To secondorder, the Taylor expansion is( )y i+1 = y i + hf i + h2 df, (27.75)2 dxx iwhere( ) ( df ∂f=dxx i∂x + f ∂f )≡ ∂f i∂yx i∂x + f ∂f ii∂y ,the last step being merely the definition of an abbreviated notation.1026