Lecture 19-20
Initial Boundary Value Problems for Parabolic PDEs
Finite-Difference Methods for Parabolic PDEs
Stability of iterations. Finite-difference methods for numerical solutions of partial differential
equations can be surprisingly inappropriate for numerical approximations. The main problem
with finite-difference methods (especially with explicit iteration schemes) is that they may
magnify the numerical round-of noise.
We can investigate instabilities of numerical finite-difference methods by employing the
separation of variables, the linear superposition principle, and the discrete Fourier transform. For
simplicity,
we
consider
the
explicit
method
uk ,l 1  1  2r  uk ,l  r  uk 1,l  uk 1,l    Fk ,l ,
(see
Lecture
17-18,
formula
(6)),
for
the
homogeneous heat equation ut  uxx  F  x, t  , (see Lecture 17-18, formula (4)), with F  x, t   0
and   t     t   0 . The explicit scheme takes the from
uk ,l 1  1  2r  uk ,l  r  uk 1,l  uk 1,l 
(1)
for k  1, 2,..., n  1 and l  0,1, 2,... , subject to the boundary conditions u0,l  un,l  0 and initial
conditions uk ,0  u0  xk  . Freeze the time level t  tl and expand the values of uk ,l in the discrete
Fourier sine-transform:
n
  kj 
uk ,l   a j ,l sin 
,
 n 
j 1
k  0,1,..., n .
(2)
The boundary conditions u0,l  un,l  0 are satisfied for any l  0. Because of the linear
superposition principle, we can consider the time iterations of each term in the sum (2)
j
separately. Hence, we can substitute uk ,l  a j ,l sin  j k  with  j 
into the explicit method
n
(1) and obtain


a j ,l 1 sin  j k   1  2r  a j ,l sin  j k   ra j ,l sin  j  k  1   sin  j  k  1  .
Because of the trigonometric formula,
sin   k  1   sin   k 1   2cos   sin  k  ,
The factor sin  j k  cancels out, and we obtain a first-order difference equation in l for fixed j :
a j ,l 1  Q j a j ,l ,
where
1
(3)
Q j  1  2r  2r cos  j  .
(4)
Because the factor Q j is l -independent, the amplitude a j ,l of the Fourier mode sin  j k  changes
during iterations in l , according to the powers of the factor Q j :
a j ,l  Qlj a j ,0 ,
l  0,1,... .
The amplitude a j ,l grows in l if Q j  1 and it is bounded or decaying if Q j  1 . Therefore, the
explicit method is stable when
Q j  1,
j  1, 2,..., n .
(5)
Because Q j  1 for r  0 in (3), the stability constraint (4) can be rewritten as follows:
 j 
1  4r sin 2 
  1,
 2n 
j  1, 2,..., n ,
(6)
1
1
. When r  , the
2
2
first unstable Fourier mode corresponds to j  n , which is responsible for a pattern of time-
which results in the conditional stability of the explicit method for 0  r 
growing space-alternative sequence of uk ,l .
We illustrate the instability of the explicit finite-difference method (1) for an example with
r  0.55 for the space interval 0,1 and the initial condition u0  sin  x  . The exact solution
u  x, t   e t sin  x  describes smooth decay of an initial condition u0  x  to the zero solution
lim u  x, t   0 . Solve the problem numerically and research on the stability. (The expected
t 
smooth decay is destroyed by the noise that grows rapidly as a result of dynamical instabilities of
the explicit method.)
Using the same analysis, we can easily prove unconditional stability of the implicit and CrankNicolson
finite-difference
methods.
Consider
the
implicit
method
1  2r  uk ,l 1  r  uk 1,l 1  uk 1,l 1   uk ,l ,
(see Lecture 17-18, formula (8)), for the homogeneous
problem:
1  2r  uk ,l 1  r  uk 1,l 1  uk 1,l 1   uk ,l .
When uk ,l  a j ,l sin  j k  is substituted for an individual Fourier made with  j 
(7)
j
n
j  1, 2,..., n , the implicit method (7) reduces to the first-order difference equation (3) with
Qj 
1
.
1  2r  2r cos  j 
2
and
(8)
For any r  0 and any j  1, 2,..., n , we have 0  Q j  1 , which implies that the implicit method is
unconditionally
stable.
Similarly,
the
Crank-Nicolson
method
2 1  r  uk ,l 1  r  uk 1,l 1  uk 1,l 1   2 1  r  uk ,l  r  uk 1,l  uk 1,l     Fk ,l  Fk ,l 1  , (see Lecture 17-
18, formula (10)), for the homogeneous problem:
2 1  r  uk ,l 1  r  uk 1,l 1  uk 1,l 1   2 1  r  uk ,l  r  uk 1,l  uk 1,l     Fk ,l  Fk ,l 1 
(9)
reduces to the first-order difference equation (3) with
Qj 
1  r  r cos  j 
1  r  r cos  j 
.
(10)
For any r  0 and any j  1, 2,..., n , we have 0  Q j  1 , which implies that the Crank-Nicolson
method is unconditionally stable.
Convergence of iterations. Analysis of convergence and stability of the finite-difference
method provides a useful tool to control accuracy of the numerical solutions. If the step size h of
the discrete spatial grid is decreased, the truncation error of the central-difference numerical
derivatives reduces to zero as O  h 2  . For instance, if h is halved, then the truncation error is
quartered. However, the stability of the explicit method depends on r 

, that is, the time step
h2
 must be adjusted accordingly, to restore the stability of the explicit iterations. This implies that
if h is halved, than  needs to be quartered, and the computational complexity increases eight
times. No adjustment of the time step  is required in the Implicit and Crank-Nicolson methods
because of their unconditional stability. The difference between these two methods is in the
convergence rate for the truncation error: O   in the implicit method versus O  2  in the
Crank-Nicolson method. To reduce the error by a factor of four when the step size h is halved,
the time step  must be quartered in the implicit method and halved in Crank-Nicolson method.
We illustrate the convergence of the three finite-difference methods for the homogeneous heat
equation on the space interval 0,1 with the initial condition u0  sin  2 x  . The exact solution
is u  x, t   e 4 t sin  2 x  . Compares the exact solution with the three numerical solutions for
2
two step size h  0.1 and h  0.05 . The time step is adjusted as   0.1h 2 in the explicit method,
and so r  0.1 is preserved. On the other hand, the time step is adjusted as   0.1h in the
implicit and Crank-Nicolson methods so that r  1 for h  0.1 and r  2 for h  0.05 . Solve the
problem numerically and research on the stability and convergence.
Nonlinear heat equation. Although the analysis of convergence and stability is much more
complicated
when
ut  uxx  F  x, t , u, ux  ,
the
a  x  b,
function
t 0,
F  x, t , u, ux  in
the
PDE
(see Lecture 17-18, formula (1)) , is nonlinear in
u , the finite-difference methods can be extended to nonlinear problems. For illustration, we
consider the nonlinear heat equation
3
ut  uxx  u 1  u  ,
 L  x  L,
t  0,
(11)
subject to the boundary conditions
u  L, t   1,
u  L, t   1 ,
(12)
And the initial condition
u  x, 0  
1
1  tanh x  .
2
(13)
The initial condition corresponds to the transition front between the equilibrium state u  1 for
large negative x and the equilibrium state u  0 for large positive x . When L large ( L  20 ), the
truncation of the solution on the bounded interval  L, L introduces a small error, which is
invisible in numerical simulations. The stability constraint on iterations of the explicit method is
chosen from the linear part of the nonlinear equation (11). Solve the problem numerically and
research on the stability and convergence for x  20, 20 and t  0,5 .
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