Implicit Methods

Abstract

In the previous chapter we studied explicit time marching procedures, i.e. methods where the unknown quantity at the new time level is computed using only quantities at the known preceeding time level. These methods offer efficient computation of each time step but usually stability constraints, rather than accuracy considerations, limit the size of the time step and therefore lead to costly simulations. In the present chapter we will investigate implicit time marching procedures. This class of methods allows the value of an unknown quantity at the new time level to depend also on values of other unknown quantities at the new time level. This eliminates or strongly reduces the restriction on the time step by the stability constraint. The computational effort per time step is however in general increased due to the following factors:

• --- A matrix equation is generated which requires the simultaneous solution of elevation and both horizontal velocity components in Equations (2.19), (2.2) and (2.3).

• --- The matrix of the equation system has to be reformulated at every time step.

• --- The matrix has to be decomposed into a lower and upper triangular matrix at every time step.

• --- The two resulting triangular equation systems have to be solved using forward solution and back substitution at every time step.