6.6. Implicit and Semi-Implicit Schemes
Consider once again the trivial linear ODE
y  10 y
(6.42)
with y(0)  1. We can modified the Euler method by evaluating the RHS at the
“new” step instead of the “old”, i.e.,
(6.43)
yk 1  yk  h  10 yk 1
This procedure is sometimes called the backward Euler method since it can be derived
using the backward-difference approximation for y'. The simplicity of the present
example allows us to solve (6.43) for yk+1 explicitly so that
yk 1 
1
yk
1  10h
(6.44)
Obviously, this kind of explicit inversion is not possible for a general nonlinear ODE.
For example, consider the equation
y  2 y 5  6 y
which is equivalent to
y  z
and
(6.45)
z  2 y 5  6 y
Applying the backward Euler method, we get
yk 1  yk  hzk 1
zk 1  zk  h  2 yk51  6 yk 1 
(6.46)
which cannot be inverted analytically.
For this reason, the scheme is also called the
implicit Euler method.
Returning to (6.42), we recall the results of §6.2.2 that the Euler scheme fails
completely for h  3/10 , which produces a rapidly diverging oscillatory solution
instead of the exponential decay of the exact solution. For convenience, the results
obtained from the two schemes are tabulized below together with the exact result:
k
0
Implicit 1
yk
1
0.25
2
3
0.0625 0.0156
Euler
1
0.5
0.25
0.125
Exact
1 0.607
0.368
0.223
Although the “implicit” result differs significantly from the exact solution, it is a great
improvememt over that of the Euler method in that at least the correct qualitative
behavior is displayed.
Another demonstration of the power of the implicit Euler method is provided by the
stiff harmonic oscillator (6.40), which is equivalent to
x y
and
y  20 y 
1
x
100
so that the backward-difference scheme gives
xk 1  xk  h yk 1
1
1 
h



yk 1  yk  h  20 yk 1 
xk 1  
xk 1 
 yk 
100
1  20h 
100



which can be inverted to give
xk 1  xk 

yk 1 
h 
h

xk 1  
 yk 
1  20h 
100

1
h


xk 
yk 

2
h
1  20h 

1
100 1  20h 
1
1  20h  xk  hyk 
1  20h  0.01h 2 
1 
h
1


 1  20h  xk  hyk  
 yk 
2 
1  20h 
100 1  20h  0.01h


1
 yk  0.01hxk 
1  20h  0.01h 2
(6.47)
If, as before, we start at t  10 and take h  0.2, a qualitatively correct solution is
obtained as shown in Fig.6.13 [cf. the Euler result in Fig.6.11].
To create a stable 2nd order implicit scheme, we can average the “old” and the “new”
on the RHS. ( This is also the basis for the semi-implicit Crank-Nicolson scheme for
solving nonlinear PDEs [see Chapter 11] ). For example, (6.42) is written as
yk 1  yk  10h 

1
 yk  yk 1 
2
(6.48)
1  5h
yk
1  5h
(6.49)
Expanding in powers of h, we have
yk 1  1  5h  1  5h  25h 2  125h 3 
 1  10h  50h 2  250h 3 
Comparing with the Taylor expansion
1
1
yk 1  yk  hyk  h 2 yk  h 3 yk
2
6
y
k
y
k
1
1
2
3
 yk  10hyk  h 2  10  yk  h 3  10  yk 
2
6
we see that (6.49) is accurate to O(h3), as claimed.
Turning to nonlinear systems, we now apply the 1st order implicit method to the fox
rabies system, first introduced in §2.2.2 and solved in MF05 using the Mathematica
NDSolve command with the RungeKutta option. To begin, we insert the constraint
X  Y  Z  N into the original coupled equations
X  aX   b   N  X   XZ
Y   XZ    b   N  Y
(2.22)
Z  Y    b   N  Z
to display the nonlinear terms more clearly, viz.,
X   a  b X   X 2   XY       XZ
Y     b Y   XZ   XY   Y 2   YZ
(6.51)
Z  Y    b Z   XZ   YZ   Z 2
To 1st order accuracy, the corresponding implicit algorithm is
X k 1  X k  h  a  b  X k 1   X k21   X k 1Yk 1       X k 1Z k 1 
Yk 1  Yk  h     b  Yk 1   X k 1Z k 1   X k 1Yk 1   Yk21   Yk 1Z k 1 
Z k 1  Z k  h  Yk 1    b  Z k 1   X k 1Z k 1   Yk 1Z k 1   Z k21 
(6.52)
The presence of the nonlinear terms prevents an analytic inversion of (6.52). To
proceed further, the standard procedure is to expand each nonlinear term according to
 f 
 f 
f  xk 1 , zk 1   f  xk , zk    xk 1  xk   
  zk 1  zk   

(6.53)
 x  xk , zk
 z  xk , zk
For example,
X k 1Zk 1   XZ k 1
X k Zk   X k 1  X k  Zk   Zk 1  Zk  X k
(6.54)
where the RHS is linear in both Xk+1 and Zk+1. Converting each nonlinear term in
(6.52) this way and collecting terms, we get, with the help of Mathematica,
1  hA1k  X k 1  hB1kYk 1  hC1k Zk 1  X k  hD1k
hA2k X k 1  1  hB2k  X k 1  hC2k Zk 1  Yk  hD2k
(6.55)
hA3k X k 1  hB3kYk 1  1  hC3k  Zk 1  Zk  hD3k
where
A1k   b  a     Nk  X k    Zk
B1k   X k
A2k   Yk   Z k
B2k    b    N k  Yk 
C1k      X k
D1k   N k X k   X k Z k
C2k   Yk   X k
A3k   Z k
B3k   Z k  
C3k    b    Nk  Zk 
D2k   N kYk   X k Z k
D3k   N k Z k
N k  X k  Yk  Z k
Since (6.55) is linear in Xk+1 , Yk+1 , and Yk+1 , it can be inverted to give the formula for
step k+1. This length code is in
Semi-Implicit Scheme Applied to Fox Rabies System
MF27.nb
MF27.mws
Taking the same initial values and parameters as in MF05, the trajectory for h  0.004
and 5000 time steps is shown in Fig.6.14, which, aside from orientation, closely
resembles that in Fig.2.7. Since the implicit algorithm is of the lowest order and no
adaptive step size is used, the computing time for MF27 is much longer than in MF05
owing to the small h required for stability.
As can be imagined, implicit and semi-implicit schemes are much more complicated
than the explicit ones. The incorporation of an adaptive step size is particularly
difficult. Also, stability of such schemes are not guaranteed, though they usually
are.
Fig.6.13.
Implicit Euler method applied to stiff equation.
fig.6.14. Semi-implicit solution of fox rabies equation.