5.4 Coupled Equations and Their Solution
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in some of the other equations. There are two types of approaches to such
problems. In the first, all variables are solved for simultaneously. In the other,
each equation is solved for its dominant variable, treating the other variables
as known, and one iterates through the equations until the solution of the
coupled system is obtained. The two approaches also may be mixed. We
call these simultaneous and sequential methods, respectively, and they are
described in more detail below.
5.4.1 Simultaneous Solution
In simultaneous methods, all the equations are considered part of a single system. The discretized equations of fluid mechanics have, after linearization,
a block-banded structure. Direct solution of these equations would be very
expensive, especially when the equations are non-linear and the problem is
three-dimensional. Iterative solution techniques for coupled systems are generalizations of methods for single equations. The methods described above
were chosen for their applicability to coupled systems. Simultaneous solution
methods based on iterative solvers have been developed by several authors;
see e.g. papers by Galpin and Raithby (1986), Deng et al. (1994), and Weiss
et al. (1999).
5.4.2 Sequential Solution
When the equations are linear and tightly coupled, the simultaneous approach
is best. However, the equations may be so complex and non-linear that coupled methods are difficult and expensive to use. It may then be preferable to
treat each equation as if it has only a single unknown, temporarily treating
the other variables as known, using the best currently available values for
them. The equations are then solved in turn, repeating the cycle until all
equations are satisfied. When using this type of method, two points need to
be borne in mind:
0 Since some terms, e.g. the coefficients and source terms that depend on
the other variables change as the computation proceeds, it is inefficient
to solve the equations accurately at each iteration. That being the case,
direct solvers are unnecessary and iterative solvers are preferred. Iterations
performed on each equation are called inner iterations.
0 In order to obtain a solution which satisfies all of the equations, the coefficient matrices and source vector must be updated after each cycle and the
process repeated. The cycles are called outer iterations.
Optimization of this type of solution method requires careful choice of the
number of inner iterations per outer iteration. It is also necessary to limit
the change in each variable from one outer iteration to the next (underrelaxation), because a change in one variable changes the coefficients in the
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in some of the other equations. There are two types of approaches to such
problems. In the first, all variables are solved for simultaneously. In the other,
each equation is solved for its dominant variable, treating the other variables
as known, and one iterates through the equations until the solution of the
coupled system is obtained. The two approaches also may be mixed. We
call these simultaneous and sequential methods, respectively, and they are
described in more detail below.
5.4.1 Simultaneous Solution
In simultaneous methods, all the equations are considered part of a single system. The discretized equations of fluid mechanics have, after linearization,
a block-banded structure. Direct solution of these equations would be very
expensive, especially when the equations are non-linear and the problem is
three-dimensional. Iterative solution techniques for coupled systems are generalizations of methods for single equations. The methods described above
were chosen for their applicability to coupled systems. Simultaneous solution
methods based on iterative solvers have been developed by several authors;
see e.g. papers by Galpin and Raithby (1986), Deng et al. (1994), and Weiss
et al. (1999).
5.4.2 Sequential Solution
When the equations are linear and tightly coupled, the simultaneous approach
is best. However, the equations may be so complex and non-linear that coupled methods are difficult and expensive to use. It may then be preferable to
treat each equation as if it has only a single unknown, temporarily treating
the other variables as known, using the best currently available values for
them. The equations are then solved in turn, repeating the cycle until all
equations are satisfied. When using this type of method, two points need to
be borne in mind:
0 Since some terms, e.g. the coefficients and source terms that depend on
the other variables change as the computation proceeds, it is inefficient
to solve the equations accurately at each iteration. That being the case,
direct solvers are unnecessary and iterative solvers are preferred. Iterations
performed on each equation are called inner iterations.
0 In order to obtain a solution which satisfies all of the equations, the coefficient matrices and source vector must be updated after each cycle and the
process repeated. The cycles are called outer iterations.
Optimization of this type of solution method requires careful choice of the
number of inner iterations per outer iteration. It is also necessary to limit
the change in each variable from one outer iteration to the next (underrelaxation), because a change in one variable changes the coefficients in the