4.5 Implementation of Boundary Conditions
81
interpolation, we find that each flux depends on 15 nodal values and the
algebraic equation for one CV involves 25 values. The solution of the resulting
equation system would be very expensive (see Chap. 5).
A way around this problem lies in the deferred-correction approach, that
will be described in Sect. 5.6.
One should bear in mind that a higher-order approximation does not necessarily guarantee a more accurate solution on any single grid; high accuracy
is achieved only when the grid is fine enough to capture all of the essential
details of the solution; a t what grid size this happens can be determined only
by systematic grid refinement.
4.4.5 Other Schemes
A large number of approximations to the convective fluxes have been proposed; it is beyond the scope of this book to discuss all of them. The approach
used above can be used to derive nearly all of them. We shall describe a few
of them briefly.
One can approximate 4, by linear extrapolation from two upstream nodes,
leading to the so called linear upwind scheme (LUDS). This scheme is of
second order accuracy but, as it is more complex than CDS and can produce
unbounded solutions, the latter is a better choice.
Another approach, proposed by Raithby (1976), is to extrapolate from
the upwind side, but along a streamline rather than a grid line (skew upwind
schemes). First- and second-order schemes corresponding to the upwind and
linear upwind schemes have been proposed. They have better accuracy than
schemes based on extrapolation along grid lines. However, these schemes are
very complex (there are many possible directions of flow) and a lot of interpolation is required. Since these schemes may produce oscillatory solutions
when the grid is not sufficiently fine and are difficult to program, they have
not found widespread use.
It is also possible to blend two or more different approximations. One
example that saw a great deal of use in the 1970s and early 1980s is the
hybrid scheme of Spalding (1972), which switches between UDS and CDS,
depending on the local value of the Peclet number. Other researchers have
proposed blending of lower and higher-order schemes to avoid unphysical
oscillations, especially for compressible flows with shocks. Some of these ideas
will be mentioned in Chap. 10. Blending rnay be used to improve the rate of
convergence of some iterative solvers, as we shall show below.
4.5 Implementation of Boundary Conditions
Each CV provides one algebraic equation. Volume integrals are calculated
in the same way for every CV, but fluxes through CV faces coinciding with
81
interpolation, we find that each flux depends on 15 nodal values and the
algebraic equation for one CV involves 25 values. The solution of the resulting
equation system would be very expensive (see Chap. 5).
A way around this problem lies in the deferred-correction approach, that
will be described in Sect. 5.6.
One should bear in mind that a higher-order approximation does not necessarily guarantee a more accurate solution on any single grid; high accuracy
is achieved only when the grid is fine enough to capture all of the essential
details of the solution; a t what grid size this happens can be determined only
by systematic grid refinement.
4.4.5 Other Schemes
A large number of approximations to the convective fluxes have been proposed; it is beyond the scope of this book to discuss all of them. The approach
used above can be used to derive nearly all of them. We shall describe a few
of them briefly.
One can approximate 4, by linear extrapolation from two upstream nodes,
leading to the so called linear upwind scheme (LUDS). This scheme is of
second order accuracy but, as it is more complex than CDS and can produce
unbounded solutions, the latter is a better choice.
Another approach, proposed by Raithby (1976), is to extrapolate from
the upwind side, but along a streamline rather than a grid line (skew upwind
schemes). First- and second-order schemes corresponding to the upwind and
linear upwind schemes have been proposed. They have better accuracy than
schemes based on extrapolation along grid lines. However, these schemes are
very complex (there are many possible directions of flow) and a lot of interpolation is required. Since these schemes may produce oscillatory solutions
when the grid is not sufficiently fine and are difficult to program, they have
not found widespread use.
It is also possible to blend two or more different approximations. One
example that saw a great deal of use in the 1970s and early 1980s is the
hybrid scheme of Spalding (1972), which switches between UDS and CDS,
depending on the local value of the Peclet number. Other researchers have
proposed blending of lower and higher-order schemes to avoid unphysical
oscillations, especially for compressible flows with shocks. Some of these ideas
will be mentioned in Chap. 10. Blending rnay be used to improve the rate of
convergence of some iterative solvers, as we shall show below.
4.5 Implementation of Boundary Conditions
Each CV provides one algebraic equation. Volume integrals are calculated
in the same way for every CV, but fluxes through CV faces coinciding with