SKEDSOFT

Implicit methods:  Explicit method has disadvantage that they have stability condition on mesh ratio parameter λ.

Explicit methods have the disadvantage that they have a stability condition on the mesh ratio parameter λ. We have seen that the Schmidt method is stable for

λ ≤ 0.5. This condition severely restricts the values that can be used for the step lengths h and k. In most practical problems, where the computation is to be done up to large value of t, these methods are not useful because the time taken is too high. In such cases, we use the implicit methods. We shall discuss the most popular and useful method called the Crank-Nicolson method. There are a number of ways of deriving this method. We describe one of the simple ways. Denote ∇t as the backward difference in the time direction. From

we write the relation

...................1.1

Now, approximate

......................1.2

If we expand the operator on the right hand side, we get

which agrees with the first two terms on the right hand side of (5.46). Applying the differential equation at the nodal point (i, j 1),

we obtain,

............................1.3

Using the approximation given in (1.2) to left hand side and the central difference approximation 

central approximation formula

to the right hand side, we obtain

or,

or,