The source and coefficients are assumed as average values taken at a time level with and are therefore specified by the superscript . They can be expressed by quantities taken at the time level and respectively. The average values in Eqn.. (40.8) may be taken as follows:

	(40.9)
	(40.10)

The theoretical limiting values for the blending factor are zero and one, respectively. However, when is set equal to zero the time average of the fluxes and sources are taken completely at the old time level resulting in an explicit scheme. Therefore, may also be chosen in the interval (0,1]. Practically, in the case of all fluxes are calculated at the new time step, resulting in a fully implicit scheme (implicit Euler scheme) with very good stability properties. There is no restriction for the time step with respect to numerical stability. The equations for steady problems result from the limit

For the Crank-Nicolson scheme, which is of second orde accuracy is obtained. For this scheme there exists a limiting value for the time step ensuring numerical stability, but due to the complexity of the underlying differential equation it is not possible to estimate this limiting value in advance.

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