# Discrete Numerical Methods in Physics and Engineering by Greenspan

By Greenspan

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81), ( 2 . 85) 1 i (2) 30 y. = I The term 1 i (2) 30 i + 1g4 y1 ) . 85) soon dominates t h e term 1 4(I)and results 30 2 in a n overflow which is due strictly t o roundoff error. Thus, we have illustrated two unstable c a l c u l a t i o n s , one of which c a n be corrected by decreasing t h e grid s i z e , t h e other of which ORDINARY DIFFERENTIAL EQUATIONS 52 cannot be so corrected. For nonlinear problems instability a n a l y s i s often is not possible, and the method of decreasing the grid size is usually the first s t e p one t a k e s i n trying t o eliminate instability which is not due t o programming or machine error.

25 , ,... 78), i = o,l,21... I and y 2 = 1i , y 3 = - '1 16 from which it follows that y n - =';I Y 4 = ~ 1' " . ' Y 0 a s n- nt1 1 n . However, suppose that in-place of doing exact calculations I as above , one duplicates what a digital computer does and allows roundoff error t o be introduced in the following simple way: round off all given data and results of a l l arithmetic operations t o one decimal place. 3 , a n d , from ( 2 . 7 7 ) , n-4 yn=2 ,1123, which r e s u l t s quickly i n overflow.

However, since symbol man- ipulation is now advancing a s a computer science discipline, the Taylor series method is returning t o a position of stature, and is described a s follows. y hntl 4 3 h i v h v t-y (x)t z y (x)t . 0 . 3! tG (nt2)(u), Let G[a,h] be a set of grid points. < x t h. 10), ORDINARY DIFFERENTIAL EQUATIONS 30 5 n t hntl y @ ) ( a )t - (ntl)! ~ ( ~ ~ " ( ae <) ,4 < a t h . 12) w e call a n nth order approximation t o y ( a t h ) , a n d , from (2. 14) 3 hn ( n t l ) h2 h iv (a) t o *t--Y y ' = y ' ( a ) t h y " ( a ) t - y " ' ( a ) t -y (a) 1 2 3!