SUPPLEMENTARY PROBLEMS
1. Rewrite the following IVPs as first order systems:
(a) y  y  0, y(0)  1, y(0)  0 ,
(b) x  x  2x  1  2t, x(0)  1, x(0)  1 ,
(c) y  2y  y  2y  1  2t, y(0)  1, y(0)  1, y(0)  0 ,
(d) y  t 2  y  z2 , z  t  z  y3 , y(0)  0, y(0)  1, z(0)  1, z(0)  0 .
Write the following IVP as an autonomous system with two components:
(e) u  12t 4 u  sin(t), u(0)   .
2. What values should the parameters a and b have so that the following LMMs are
consistent:
(a) y n  2  ay n 1  2y n  hbf n ,
(b) y n  2  y n 1  ay n  h(f n  2  bf n ) .
3. Determine the order and error constant of the following LMMs:
(a) 3yn 2  4yn 1  yn  2hf n 2 ,
(b) yn 2  yn 1  h( 125 f n 2  23 f n 1  121 f n ) .
What value of  produces the highest order:
(c) yn 2  yn  h(f n 1  (2  )f n ) .
4. Evaluate the order of accuracy of the Verlet algorithm yn 1  2yn  yn 1  h 2f n for the
second order ODE y  f .
5. Investigate the consistency and zero-stability of the two-step LMMs:
(a) y n  2  4y n 1  3y n  2hf n ,
(b) 3yn 2  4yn 1  yn  ahf n , a  R ,
(c) y n 3  y n  2  y n 1  y n  h(f n 3  f n  2  f n 1  f n ) ,
(d) yn 2  (b 1)yn 1  byn  14 h((b  3)f n 2  (3b  1)f n ), b R ,
6. Show that the region of absolute stability of the backward Euler method u n 1  u n  hf n 1
is given by 1  k  1 . By writing k  x  iy , show that this corresponds to the exterior of
the circle (x  1)2  y2  1 .
7. Find the interval (on the real k  h axis) of absolute instability of the LMMs:
(a) y n  2  y n  12 h( f n1  3f n ) ,
(b) y n  2  y n 1  12 h(3f n1  f n ) .
8. What condition must the step length h satisfy in order to achieve absolute stability when
Euler’s method is applied to the following systems:
(a) u (t )  v(t ) , v(t )  200u(t )  20v(t ) .
 1 1 
(b) u  
u.
 1 1
(c) u  8(u  v), v   18 (v  5), u(0)  100, v(0)  20.
9. The scalar ODE y(t )  f (t, y(t )) with y(0)  y 0 may be written in the form of two
autonomous ODEs Y(t )  F(Y(t )) , where
u ( t )
Y( t )  
,
 v( t ) 
 1 
F(Y( t ))  

f (u, v)
with u(0)  0 , v(0)  y 0 . Show that the condition c j  i 1 a i, j arises naturally when we
s
ask for a RK method to produce the same numerical approximation when applied to either
version of the problem.
10. Show that the RK method applied to the IVP, y  1, y(0)  0, will not converge unless
s
b
i 1
i
 1.
11. Show that the following RK methods are consistent and find their order of accuracy:
(a) yn 1  yn  12 h(k1  k 2 ), k1  f (t n , yn ), k 2  f (t n  h, yn  hk1 ) .
(b) yn 1  yn  13 h(2k1  k 2 ), k1  f (t n , yn ), k 2  f (t n  23 h, yn  23 hk1 ) .