2.5 Properties of Numerical Solution Methods
31
2.4.7 Convergence Criteria
Finally, one needs to set the convergence criteria for the iterative method.
Usually, there are two levels of iterations: inner iterations, within which the
linear equation are solved, and outer iterations, that deal with the nonlinearity and coupling of the equations. Deciding when to stop the iterative
process on each level is important, from both the accuracy and efficiency
points of view. These issues are dealt with in Chaps. 5 and 11.
2.5 Properties of Numerical Solution Methods
The solution method should have certain properties. In most cases, it is not
possible to analyze the complete solution method. One analyzes the components of the method; if the components do not possess the desired properties,
neither will the complete method but the reverse is not necessarily true. The
most important properties are summarized below.
2.5.1 Consistency
The discretization should become exact as the grid spacing tends to zero. The
difference between the discretized equation and the exact one is called the
truncation error. It is usually estimated by replacing all the nodal values in
the discrete approximation by a Taylor series expansion about a single point.
As a result one recovers the original differential equation plus a remainder,
which represents the truncation error. For a method to be consistent, the
truncation error must become zero when the mesh spacing At + 0 and/or
Axi + 0. Truncation error is usually proportional to a power of the grid
spacing Axi and/or the time step At. If the most important term is proportional to AX)^ or (At)n we call the method an nth-order approximation;
n > 0 is required for consistency. Ideally, all terms should be discretized with
approximations of the same order of accuracy; however, some terms (e.g.
convective terms in high Reynolds number flows or diffusive terms in low
Reynolds number flows) may be dominant in a particular flow and it may be
reasonable to treat them with more accuracy than the others.
Some discretization methods lead to truncation errors which are functions
of the ratio of Axi to At or vice versa. In such a case the consistency requirement is only conditionally fulfilled: Axi and At must be reduced in a way
that allows the appropriate ratio to go to zero. In the next two chapters we
shall demonstrate consistency for several discretization schemes.
Even if the approximations are consistent, it does not necessarily mean
that the solution of the discretized equation system will become the exact
solution of the differential equation in the limit of small step size. For this to
happen, the solution method has to be stable; this is defined below.
31
2.4.7 Convergence Criteria
Finally, one needs to set the convergence criteria for the iterative method.
Usually, there are two levels of iterations: inner iterations, within which the
linear equation are solved, and outer iterations, that deal with the nonlinearity and coupling of the equations. Deciding when to stop the iterative
process on each level is important, from both the accuracy and efficiency
points of view. These issues are dealt with in Chaps. 5 and 11.
2.5 Properties of Numerical Solution Methods
The solution method should have certain properties. In most cases, it is not
possible to analyze the complete solution method. One analyzes the components of the method; if the components do not possess the desired properties,
neither will the complete method but the reverse is not necessarily true. The
most important properties are summarized below.
2.5.1 Consistency
The discretization should become exact as the grid spacing tends to zero. The
difference between the discretized equation and the exact one is called the
truncation error. It is usually estimated by replacing all the nodal values in
the discrete approximation by a Taylor series expansion about a single point.
As a result one recovers the original differential equation plus a remainder,
which represents the truncation error. For a method to be consistent, the
truncation error must become zero when the mesh spacing At + 0 and/or
Axi + 0. Truncation error is usually proportional to a power of the grid
spacing Axi and/or the time step At. If the most important term is proportional to AX)^ or (At)n we call the method an nth-order approximation;
n > 0 is required for consistency. Ideally, all terms should be discretized with
approximations of the same order of accuracy; however, some terms (e.g.
convective terms in high Reynolds number flows or diffusive terms in low
Reynolds number flows) may be dominant in a particular flow and it may be
reasonable to treat them with more accuracy than the others.
Some discretization methods lead to truncation errors which are functions
of the ratio of Axi to At or vice versa. In such a case the consistency requirement is only conditionally fulfilled: Axi and At must be reduced in a way
that allows the appropriate ratio to go to zero. In the next two chapters we
shall demonstrate consistency for several discretization schemes.
Even if the approximations are consistent, it does not necessarily mean
that the solution of the discretized equation system will become the exact
solution of the differential equation in the limit of small step size. For this to
happen, the solution method has to be stable; this is defined below.