2.5 Properties of Numerical Solution Methods
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2.5.4 Conservation
Since the equations to be solved are conservation laws, the numerical scheme
should also - on both a local and a global basis - respect these laws. This
means that, a t steady state and in the absence of sources, the amount of a
conserved quantity leaving a closed volume is equal to the amount entering
that volume. If the strong conservation form of equations and a finite volume
method are used, this is guaranteed for each individual control volume and
for the solution domain as a whole. Other discretization methods can be made
conservative if care is taken in the choice of approximations. The treatment
of sources or sink terms should be consistent so that the total source or sink
in the domain is equal to the net flux of the conserved quantity through the
boundaries.
This is an important property of the solution method, since it imposes a
constraint on the solution error. If the conservation of mass, momentum and
energy are insured, the error can only improperly distribute these quantities
over the solution domain. Non-conservative schemes can produce artificial
sources and sinks, changing the balance both locally and globally. However,
non-conservative schemes can be consistent and stable and therefore lead
to correct solutions in the limit of very fine grids. The errors due to nonconservation are in most cases appreciable only on relatively coarse grids.
The problem is that it is difficult to know on which grid are these errors
small enough. Conservative schemes are therefore preferred.
2.5.5 Boundedness
Numerical solutions should lie within proper bounds. Physically non-negative
quantities (like density, kinetic energy of turbulence) must always be positive;
other quantities, such as concentration, must lie between 0% and 100%. In
the absence of sources, some equations (e.g. the heat equation for the temperature when no heat sources are present) require that the minimum and
maximum values of the variable be found on the boundaries of the domain.
These conditions should be inherited by the numerical approximation.
Boundedness is difficult to guarantee. We shall show later on that only
some first order schemes guarantee this property. All higher-order schemes
can produce unbounded solutions; fortunately, this usually happens only on
grids that are too coarse, so a solution with undershoots and overshoots
is usually an indication that the errors in the solution are large and the
grid needs some refinement (at least locally). The problem is that schemes
prone to producing unbounded solutions may have stability and convergence
problems. These methods should be avoided, if possible.
2.5.6 Realizability
Models of phenomena which are too complex to treat directly (for example,
turbulence, combustion, or multiphase flow) should be designed to guarantee
33
2.5.4 Conservation
Since the equations to be solved are conservation laws, the numerical scheme
should also - on both a local and a global basis - respect these laws. This
means that, a t steady state and in the absence of sources, the amount of a
conserved quantity leaving a closed volume is equal to the amount entering
that volume. If the strong conservation form of equations and a finite volume
method are used, this is guaranteed for each individual control volume and
for the solution domain as a whole. Other discretization methods can be made
conservative if care is taken in the choice of approximations. The treatment
of sources or sink terms should be consistent so that the total source or sink
in the domain is equal to the net flux of the conserved quantity through the
boundaries.
This is an important property of the solution method, since it imposes a
constraint on the solution error. If the conservation of mass, momentum and
energy are insured, the error can only improperly distribute these quantities
over the solution domain. Non-conservative schemes can produce artificial
sources and sinks, changing the balance both locally and globally. However,
non-conservative schemes can be consistent and stable and therefore lead
to correct solutions in the limit of very fine grids. The errors due to nonconservation are in most cases appreciable only on relatively coarse grids.
The problem is that it is difficult to know on which grid are these errors
small enough. Conservative schemes are therefore preferred.
2.5.5 Boundedness
Numerical solutions should lie within proper bounds. Physically non-negative
quantities (like density, kinetic energy of turbulence) must always be positive;
other quantities, such as concentration, must lie between 0% and 100%. In
the absence of sources, some equations (e.g. the heat equation for the temperature when no heat sources are present) require that the minimum and
maximum values of the variable be found on the boundaries of the domain.
These conditions should be inherited by the numerical approximation.
Boundedness is difficult to guarantee. We shall show later on that only
some first order schemes guarantee this property. All higher-order schemes
can produce unbounded solutions; fortunately, this usually happens only on
grids that are too coarse, so a solution with undershoots and overshoots
is usually an indication that the errors in the solution are large and the
grid needs some refinement (at least locally). The problem is that schemes
prone to producing unbounded solutions may have stability and convergence
problems. These methods should be avoided, if possible.
2.5.6 Realizability
Models of phenomena which are too complex to treat directly (for example,
turbulence, combustion, or multiphase flow) should be designed to guarantee