Implicit Methods Wave Equations

Implicit method : Explicit method has disadvantage that they have stability condition on mesh ratio parameter λ.sp to overcome this disadvantage this method is used.

As we know Explicit methods have the disadvantage that they have a stability condition on the mesh ratio parameter r = (ck)/h. Explicit methods are stable for

r ≤ 1.0. This condition 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 a large value of t, these methods are not useful because the time consumed is too high. In such cases, we use the implicit methods. We derive the following two implicit methods.

(i) We write the following approximations at (x_i, t_j).

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

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

Hence, the difference approximation to the wave equation at the node (x_i , t_j) is given by

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

or,

..........................1.4

where r = (kc/h). We can expand the central differences and write

We get

The nodal points that are used in the method are given in Figure

figure : Nodes in implicit method