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10. Compressible Flow
equivalents in finite volume methods), many modern methods for compressible flow use central differences everywhere except near the discontinuities
where special upwind methods are applied.
These are the issues that must be faced in the design of numerical methods
for compressible flows. We now look, in a very general and superficial way,
a t some of the methods that have been proposed to deal with them.
10.3.1 An Overview of Some Specific Methods
The earliest schemes were based on explicit methods and central differencing.
One of the most notable of these is the method of MacCormack (1969) which
is still used. To avoid the problem of oscillations at shocks in this type of
method, it is necessary to introduce artificial dissipation into the equations.
The usual second-order dissipation (equivalent to ordinary viscosity) would
smooth the solution everywhere so a term more sensitive to the rapid variation at the shock is needed. A fourth-order dissipative term i.e., an added
term that contains fourth derivatives of the velocity, is the most common
addition but higher-order terms have also been used.
The first effective implicit methods were developed by Beam and Warming
(1978). Their method is based on approximate factorization of the CrankNicolson method and can be considered an extension of the AD1 method
presented in Chap. 6 t o compressible flow. As with the AD1 method, this
method has an optimum time step for convergence t o a steady solution. The
use of central differences again requires addition of an explicit fourth-order
dissipative term to the equations.
More recently, there has been an interest in upwind schemes of greater
sophistication. The objective is always to produce a well-defined discontinuity without introducing an undue amount of error into the smooth part of
the solution. One scheme for accomplishing this is the flux-vector-splitting
method of Steger and Warming (1981) to which a number of modifications
and extensions have been suggested. The idea is to locally split the flux (since
the application is t o the Euler equations, this means the convective flux of
momentum) into components that flow along the various characteristics of
the equations. In general, these fluxes flow in different directions. Each flux
is then treated by an upwind method appropriate to the direction in which
it flows. The resulting method is fairly complex but the upwinding provides
stability and smoothness at discontinuities.
Finally, we mention a class of schemes that use limiters to provide smooth
and accurate solutions. The earliest of these (and one of the easiest to explain)
is the flux-corrected transport (FCT) method of Boris and Book (1973). In a
one-dimensional version of the method, one might compute the solution using
a simple first-order upwind method. The diffusive error in the solution can be
estimated (one way is to use a higher-order scheme and take the difference).
This estimated error is then subtracted from the solution (a so-called antidiffusive step) but only to the extent that it does not produce oscillations.
10. Compressible Flow
equivalents in finite volume methods), many modern methods for compressible flow use central differences everywhere except near the discontinuities
where special upwind methods are applied.
These are the issues that must be faced in the design of numerical methods
for compressible flows. We now look, in a very general and superficial way,
a t some of the methods that have been proposed to deal with them.
10.3.1 An Overview of Some Specific Methods
The earliest schemes were based on explicit methods and central differencing.
One of the most notable of these is the method of MacCormack (1969) which
is still used. To avoid the problem of oscillations at shocks in this type of
method, it is necessary to introduce artificial dissipation into the equations.
The usual second-order dissipation (equivalent to ordinary viscosity) would
smooth the solution everywhere so a term more sensitive to the rapid variation at the shock is needed. A fourth-order dissipative term i.e., an added
term that contains fourth derivatives of the velocity, is the most common
addition but higher-order terms have also been used.
The first effective implicit methods were developed by Beam and Warming
(1978). Their method is based on approximate factorization of the CrankNicolson method and can be considered an extension of the AD1 method
presented in Chap. 6 t o compressible flow. As with the AD1 method, this
method has an optimum time step for convergence t o a steady solution. The
use of central differences again requires addition of an explicit fourth-order
dissipative term to the equations.
More recently, there has been an interest in upwind schemes of greater
sophistication. The objective is always to produce a well-defined discontinuity without introducing an undue amount of error into the smooth part of
the solution. One scheme for accomplishing this is the flux-vector-splitting
method of Steger and Warming (1981) to which a number of modifications
and extensions have been suggested. The idea is to locally split the flux (since
the application is t o the Euler equations, this means the convective flux of
momentum) into components that flow along the various characteristics of
the equations. In general, these fluxes flow in different directions. Each flux
is then treated by an upwind method appropriate to the direction in which
it flows. The resulting method is fairly complex but the upwinding provides
stability and smoothness at discontinuities.
Finally, we mention a class of schemes that use limiters to provide smooth
and accurate solutions. The earliest of these (and one of the easiest to explain)
is the flux-corrected transport (FCT) method of Boris and Book (1973). In a
one-dimensional version of the method, one might compute the solution using
a simple first-order upwind method. The diffusive error in the solution can be
estimated (one way is to use a higher-order scheme and take the difference).
This estimated error is then subtracted from the solution (a so-called antidiffusive step) but only to the extent that it does not produce oscillations.