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L β αn αn-1 1 5/12 8/12 -1/12 2 9/24 19/24 -5/24 1/24 3 251/ 646/720 -264/720 106/720 αn-2 αn-3 truncation error O(∆t3) O(∆t4) -19/720 O(∆t5) 720 m = 1, l = 0 Highest accuracy (2nd order) is obtained for β = 0 , which gives the leapfrog scheme. m = 1 , l = 1 (Milne corrector1) Eq. 14) β + αn + αn – 1 = 1 . 15) where The truncation error is 2 3 4 ∆t ∆t 1 ε = ∆tq′′ ( – β + α n – 1 ) + -------- q′′′  --- – β – α n – 1 + -------- q′′′′ ( – β + α n – 1 ) + O ( ∆t ) . 3  3! 2! 4 1 1 From this we can see that β = --- , α n = --- , α n – 1 = --- gives fourth-order accuracy.

It becomes intractable for complex schemes. The second method, the energy method, is more widely applicable, even for some nonlinear equations. 75). ” j Here the summation obviously must be taken over a finite number of grid points. This is not an important limitation because in practice we are always dealing with a finite n number of grid points. If the sum is bounded, then each u j must also be bounded. n+1 Whereas in the direct method we checked max ( j ) u j check n 2 ∑ ( uj ) , here in the energy method we .

The k ’s have to be evaluated successively, which means that the function f has to be evaluated four times to take one time step. None of these f ’s can be “re-used” on the next time step. For this reason, the scheme is not very practical unless a long time step can be used. Fig. 2 provides a simple fortran example to illustrate more clearly how the fourthorder Runge-Kutta scheme actually works. The appendix to this chapter provides a proof that the scheme really has fourth-order accuracy. 4 Finite-difference schemes applied to the oscillation equation In order to test the stability of the finite-difference schemes discussed in the previous section, we must specify the form of the function f ( q, t ) .

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