Information Technology Reference
In-Depth Information
The algorithm for solving the discrete version (7.115)of(
7.82
) can be written as
slight modifications of Algorithms
7.1
and
7.3
, depending on the choice of boundary
conditions.
7.5
Implicit Numerical Methods
The numerical method for diffusion problems from Sect.
7.4
is explicit, meaning
that we can compute the value of
u
.x; t / at a space-time grid point from a formula
that is simple to evaluate. In a program, we run through all grid points at all time
levels and compute a new
u
value at each grid point by evaluating the formula.
In many other PDE problems, the values of the unknown function at a new time
level are coupled to each other. This yields a linear system of algebraic equations to
be solved instead of just evaluating a simple explicit formula. Methods where new
values are coupled in a linear system are known as
implicit
methods.
Since explicit methods are much simpler to work with than implicit methods,
one can ask why it is not sufficient to always use explicit methods. The answer to
this question lies in the instability problem we encountered with the explicit scheme
in Sect.
7.4.5
: The maximum time step is t
D
x
2
=.2k/. Say you compute the
solution with some value of x. Suppose, then, that you need a four times finer res-
olution in space. Now the time step must be reduced by a factor of 16. To simulate
from time zero to some finite time T the overall work increases by a factor of 64. If
we could stick to the old value of the time step, the increase in the work would be
by a factor of 4 only. (This reasoning may be too simplistic, because increased accu-
racy in space should normally be accompanied by a smaller time step for increased
accuracy in time. Otherwise the total error may be dominated by the temporal error
and refinements in space are just a waste of resources.)
One can formulate implicit methods for diffusion equations where there are no
stability restrictions on the time step, i.e., any t will work. Of course, a large t
yields inaccurate results, but the solution might exhibit at least correct qualitative
features in cases where an explicit scheme produces completely useless results. In
particular, if we seek the stationary solution of a diffusion problem as t
!1
and
u
t
!
0, one giant time step with an implicit method may be enough to arrive at an
accurate solution (!).
7.5.1
The Backward Euler Scheme
Equation (7.87) is our starting point for all finite difference schemes, i.e., we sample
the PDE at a space-time point .x
i
;t
`
/. The next step is to replace derivatives by
finite differences. This time we use a
backward
difference for the time derivative,
