150
6. Methods for Unsteady Problems
The scheme is implicit; the contributions from fluxes and sources at the new
time level give rise to a coupled set of equations similar to those of implicit
Euler scheme. The above equation can be written as:
where:
The term Q! represents an "additional" source term, which contains the contribution from the previous time level; it remains constant during iterations
at the new time level. The equation may also contain a source term dependent
on the new solution, so the above term needs to be stored separately.
This scheme requires very little more computational effort per step than
the first order implicit Euler scheme. Von Neumann stability analysis shows
that the scheme is unconditionally stable, but oscillatory solutions (and even
instability) are possible for large time steps. This may be attributed to the
possibility of the coefficient of $1-' becoming negative at large At, but is
guaranteed to be positive if At < p ( A ~ ) ~ / r ,
which is twice the maximum step
size allowed by the explicit Euler method. In practice, much larger time steps
can be used without producing oscillations; the limit is problem dependent.
This scheme can be regarded as an equal blend of first order explicit
and implicit Euler schemes. Only for equal blending is second order accuracy
obtained; for other blending factors, which may vary in space and time, the
method remains first order accurate. The stability is increased if the implicit
contribution is increased, but the accuracy is reduced.
Three Time Level Method. A fully implicit scheme of second order accuracy can be obtained by using a quadratic backward approximation in time,
as described in Sect. 6.2.4. For the ID generic
discretization in space we obtain:
transport equation and CDS
The resulting algebraic equation can be written:
6. Methods for Unsteady Problems
The scheme is implicit; the contributions from fluxes and sources at the new
time level give rise to a coupled set of equations similar to those of implicit
Euler scheme. The above equation can be written as:
where:
The term Q! represents an "additional" source term, which contains the contribution from the previous time level; it remains constant during iterations
at the new time level. The equation may also contain a source term dependent
on the new solution, so the above term needs to be stored separately.
This scheme requires very little more computational effort per step than
the first order implicit Euler scheme. Von Neumann stability analysis shows
that the scheme is unconditionally stable, but oscillatory solutions (and even
instability) are possible for large time steps. This may be attributed to the
possibility of the coefficient of $1-' becoming negative at large At, but is
guaranteed to be positive if At < p ( A ~ ) ~ / r ,
which is twice the maximum step
size allowed by the explicit Euler method. In practice, much larger time steps
can be used without producing oscillations; the limit is problem dependent.
This scheme can be regarded as an equal blend of first order explicit
and implicit Euler schemes. Only for equal blending is second order accuracy
obtained; for other blending factors, which may vary in space and time, the
method remains first order accurate. The stability is increased if the implicit
contribution is increased, but the accuracy is reduced.
Three Time Level Method. A fully implicit scheme of second order accuracy can be obtained by using a quadratic backward approximation in time,
as described in Sect. 6.2.4. For the ID generic
discretization in space we obtain:
transport equation and CDS
The resulting algebraic equation can be written: