This document discusses various numerical methods for solving systems of ordinary differential equations (ODEs), including: - Explicit methods like Euler's method can be directly applied to systems of ODEs but may require a very small time-step for stability. Implicit methods require solving a nonlinear system at each step. - Predictor-corrector methods like Heun's method or Adams-Bashforth/Adams-Moulton methods combine explicit and implicit steps to gain accuracy while maintaining stability. - Higher order ODEs can be converted to a system of first order ODEs to apply the same methods, with initial value problems (IVPs) readily solved this way but boundary value problems (

1. • The analytical solution is bounded for all
negative λr
• Stability Region
Linear Stability Analysis
 
h
i
h
n i
r
e
y
y 
 
 0
λrh
λih

2. • The stability region is shown below: a circle of
radius 1, centered at (-1,0)
• For real negative values of λ, the condition is
|λh|≤2
Linear Stability Analysis: Euler Forward
λrh
λih
-2

3. • The stability region is shown below: outside a
circle of radius 1, centered at (-1,0)
• For real negative values of λ, the method is
unconditionally stable
Linear Stability Analysis: Euler Backward
λrh
λih
2

4. • For Trapezoidal method
• The stability region is, therefore,given by
• Which implies λrh≤0
• Same as that for the exact solution.
• Unconditionally stable, does not give bounded
solution when the exact is not bounded!
Linear Stability Analysis: Trapezoidal method
   
2
/
2
/
1
2
/
2
/
1
2
,
,
1
1
1
1
h
i
h
h
i
h
y
y
y
t
f
y
t
f
h
y
y
i
r
i
r
n
n
n
n
n
n
n
n












 



1
2
/
2
/
1
2
/
2
/
1





h
i
h
h
i
h
i
r
i
r





5. • For 2nd order R-K method (Heun’s)
• The stability region is, therefore,the region
inside the shape whose boundary is given by
• Or:
Linear Stability Analysis: R-K method
   
 
 
2
/
1
2
,
,
,
2
2
1
1
1
h
h
y
y
y
t
hf
y
t
f
y
t
f
h
y
y
n
n
n
n
n
n
n
n
n
n

 










1
2
/
1 2
2


 h
h 

)
2
(
2
)
2
( h
h
h
h
h r
r
r
r
i 



 







6. -3
-2
-1
0
1
2
3
-3 -2 -1 0 1 2 3
• The stability region is shown below: centered
at (-1,0), major axis = 2√3, minor =2
• For real negative values of λ, the condition is
|λh|≤2
Stability Region for Heun’s method
λrh
λih
√3

7. • Implicit methods are stable but require
solution of a nonlinear equation at each
step
• Explicit methods require less
computational effort per step but may
need a very small time-step for stability
• Avoid the nonlinear equation solution, by
predicting the “unknown” value using
explicit method and then correcting it
using implicit
Predictor-Corrector methods

8. • For example, Heun’s method:
Predictor:
Corrector:
• Why stop at one step only? Iterate using
the corrected value in the implicit step.
• Repeat till convergence
Predictor-Corrector methods
 
n
n
n
p
n y
t
hf
y
y ,
1 


   
 
p
n
n
n
n
n
c
n y
t
f
y
t
f
h
y
y 1
1
1 ,
,
2


 


 
n
n
n
n y
t
hf
y
y ,
)
0
(
1 


   
 
)
1
(
1
1
)
(
1 ,
,
2



 

 i
n
n
n
n
n
i
n y
t
f
y
t
f
h
y
y

9. • Milne’s method (multi-step):
• Non-self starting
• Uses Simpson’s 1/3 methodology
• Predictor: interpolate a quadratic using n-2, n-
1, and n; integrate over n-3 to n+1
• Corrector: interpolate a quadratic using n-1, n,
and n+1; integrate over n-1 to n+1
Predictor-Corrector : Milne’s method

10. f
t
• Approximate f by a quadratic function:
• Integrate from -3h to h:
Milne’s method: Predictor
 
  
 
  
  
   n
n
n
f
h
h
h
t
h
t
f
h
h
h
t
h
t
f
h
h
h
t
h
t
f
2
2
2
2
2
2
1
2















-2h -h 0 h
-3h
 
n
n
n
n
h
h
n
n f
f
f
h
y
fdt
y
y 2
2
3
4
1
2
3
3
3
)
0
(
1 




 




 

11. • Approximate f by a quadratic function:
• Integrate from -h to h:
Milne’s method: Corrector
 
  
  
  
 
  
)
1
(
1
1
2
2













i
n
n
n
f
h
h
t
h
t
f
h
h
h
t
h
t
f
h
h
h
t
t
f
-2h -h 0 h
-3h
f
t
 
 
)
1
(
1
1
1
1
)
(
1 ,
4
3





 


 i
n
n
n
n
n
i
n y
t
f
f
f
h
y
y

12. • Adams method:
• Uses Adams-Bashforth (explicit) and
Adams-Moulton (implicit)
• For Example, take the 4th order method
• Predictor: interpolate a cubic using n-3, n-2, n-
1, and n; integrate over n to n+1
• Corrector: interpolate a cubic using n-2, n-1,
n, and n+1; integrate over n to n+1
Predictor-Corrector : Adams method

13. • Approximate f by a cubic function:
• Integrate from 0 to h:
Adams method: Predictor
  
   
  
   
  
   
   
    n
n
n
n
f
h
h
h
h
t
h
t
h
t
f
h
h
h
t
h
t
h
t
f
h
h
h
t
h
t
h
t
f
h
h
h
t
h
t
h
t
f
2
3
2
3
2
2
3
2
3
3
2
2
1
2
3























-2h -h 0 h
-3h













 


  n
n
n
n
n
h
n
n f
f
f
f
h
y
fdt
y
y
24
55
24
59
24
37
8
3
1
2
3
0
)
0
(
1
f
t

14. • Approximate f by a cubic function:
• Integrate from 0 to h:
Adams method: Corrector
   
   
   
   
  
   
  
   
)
1
(
1
1
2
2
3
2
2
)
(
2
2
2
3
2























i
n
n
n
n
f
h
h
h
t
h
t
h
t
f
h
h
h
h
t
h
t
h
t
f
h
h
h
h
t
t
h
t
f
h
h
h
h
t
t
h
t
f
-2h -h 0 h
-3h
f
t










 




)
1
(
1
1
2
)
(
1
8
3
24
19
24
5
24
1 i
n
n
n
n
n
i
n f
f
f
f
h
y
y

15. • If we have several dependent variables,
yi, i from 1 to m
• Derivatives could be functions of time
and one or more ys
• Initial conditions on all ys should be given
• The system may be expressed as
System of ODEs
)
,...,
,
,
(
...
)
,...,
,
,
(
)
,...,
,
,
(
2
1
2
1
2
2
2
1
1
1
m
m
m
m
m
y
y
y
t
f
dt
dy
y
y
y
t
f
dt
dy
y
y
y
t
f
dt
dy



      0
,
0
0
,
2
0
2
0
,
1
0
1 ;...;
; m
t
m
t
t
y
y
y
y
y
y 

 



16. • If we have a higher order ODE, it could be
converted into a system of ODEs
• For example,
• Could be expressed as (using y1=y and
y2=dy/dt):
Higher order ODEs
2
2
1
1
0
2
1
2
2
2
2
1
1
1
)
(
)
,
,
(
)
,
,
(
c
y
c
y
c
t
f
y
y
t
f
dt
dy
y
y
y
t
f
dt
dy






)
(
0
1
2
2
2 t
f
y
c
dt
dy
c
dt
y
d
c 



17. • The only problem is with the boundary
conditions
• There are two boundary conditions on y
• If both are specified at t=“0” (e.g., y0 and
dy/dt0): Initial Value Problem (IVP)
• If these are specified at different points
(e.g., y0 and yT): Boundary Value Problem
(BVP)
• Problems discussed till now were IVPs
Higher order ODEs

18. • The higher order IVP is readily convertible
into a system of IVPs
• The BVPs require different technique and
will be discussed later
• For now, we will look at only a system of
IVPs, and will not consider higher-order
IVPs separately, since these are
equivalent!
Higher order ODEs

19. • All the methods described earlier for a
single ODE, are applicable for a system
• Explicit methods pose no problem
• Implicit methods require the solution of a
nonlinear system of algebraic equations
• Vector notation is used to write
• where,
System of ODEs
       
0
0
with y
y
f
dt
y
d
t 
 
           T
0
,
0
,
2
0
,
1
0
T
2
1
T
2
1 ,...,
,
;
,...,
,
;
,...,
, m
m
m y
y
y
y
f
f
f
f
y
y
y
y 



20. • Euler Forward method gives:
• Or, in expanded form:
• Similarly, for other explicit methods
System of ODEs: Euler Forward
     n
n
n f
h
y
y 

1
)
,...,
,
,
(
...
)
,...,
,
,
(
)
,...,
,
,
(
,
,
2
,
1
,
1
,
,
,
2
,
1
2
,
2
1
,
2
,
,
2
,
1
1
,
1
1
,
1
n
m
n
n
n
m
n
m
n
m
n
m
n
n
n
n
n
n
m
n
n
n
n
n
y
y
y
t
f
y
y
y
y
y
t
f
y
y
y
y
y
t
f
y
y










21. • For the 4th order R-K method:
• The slopes are given by:
System of ODEs: 4th order R-K
           
 
n
n
n
n
n
n k
k
k
k
h
y
y 4
3
2
1
1
6






     
       
       
        )
,
(
4
)
2
/
,
2
/
(
3
)
2
/
,
2
/
(
2
)
,
(
1
3
2
1
h
k
y
h
t
n
h
k
y
h
t
n
h
k
y
h
t
n
y
t
n
n
n
n
n
n
n
n
n
n
n
n
f
k
f
k
f
k
f
k











22. • Euler Backward method results in:
• Or, in expanded form:
• If the fs are linear in y, a set of linear
algebraic equations
System of ODEs: Euler Backward
      1
1 
 
 n
n
n f
h
y
y
)
,...,
,
,
(
...
)
,...,
,
,
(
)
,...,
,
,
(
1
,
1
,
2
1
,
1
1
,
1
,
1
,
1
,
2
1
,
1
1
2
,
2
1
,
2
1
,
1
,
2
1
,
1
1
1
,
1
1
,
1





















n
m
n
n
n
m
n
m
n
m
n
m
n
n
n
n
n
n
m
n
n
n
n
n
y
y
y
t
f
y
y
y
y
y
t
f
y
y
y
y
y
t
f
y
y