330
11. Efficiency and Accuracy Improvement
for various reasons. Often what should be an infinite solution domain is taken
as finite and artificial boundary conditions are applied. We often have to make
assumptions about the flow a t the inlet to the solution domain as well as at
the lateral and outlet boundaries. Thus, even when the governing equations
are exact, approximations made at the boundaries may affect the solution.
Finally, the geometry may be difficult to represent exactly; often we have
to neglect details for which it is difficult to generate grids. Many codes that
use structured or block-structured grids cannot be applied to very complicated problems without simplifying the geometry.
Thus, even if we were able to solve the equations and specified boundary
conditions exactly, the result will not describe the flow exactly due t o the
errors in the model assumptions. We therefore define the modeling error as the
difference between the real flow and the exact solution of the mathematical
model.
Discretization Errors. Furthermore, we are seldom able to solve the governing equations exactly. Every numerical method produces approximate solutions, since various approximations have to be made to obtain an algebraic
system of equations that can be solved on computer. For example, in FV
methods one has to employ appropriate approximations for surface and volume integrals, variable values a t intermediate locations, and time integrals.
Obviously, the smaller the spatial and temporal discrete elements, the more
accurate these approximations become. Using better approximations can also
increase the accuracy; however, this is not a trivial matter as more accurate
approximations are more difficult to program, need more computing time
and storage, and may be difficult to apply to complex geometry. Usually, one
selects the approximations prior to writing a code so the spatial and temporal grid resolution are the only parameters a t user's disposal to control the
accuracy.
The same approximation may be very accurate in one part of the flow but
inaccurate elsewhere. Uniform spacing (either in space or in time) is seldom
optimal, since the flow may vary strongly locally in both space and time;
where the changes in variables are small, the errors will also be small. Thus,
with the same number of discrete elements and the same approximations, the
errors in the results may differ by an order of magnitude or more. Since the
computational effort is proportional to the number of discrete elements, their
proper distribution and size is essential for computational efficiency (the cost
of achieving the prescribed accuracy).
We define the discretization error as the difference between the exact
solution of the governing equations and the exact solution of the discrete
approximation.
Iteration Errors. The discretization process normally produces a coupled
set of non-linear algebraic equations. These are usually linearized and the linearized equations are also solved by an iterative method since direct solution
is usually too expensive.
11. Efficiency and Accuracy Improvement
for various reasons. Often what should be an infinite solution domain is taken
as finite and artificial boundary conditions are applied. We often have to make
assumptions about the flow a t the inlet to the solution domain as well as at
the lateral and outlet boundaries. Thus, even when the governing equations
are exact, approximations made at the boundaries may affect the solution.
Finally, the geometry may be difficult to represent exactly; often we have
to neglect details for which it is difficult to generate grids. Many codes that
use structured or block-structured grids cannot be applied to very complicated problems without simplifying the geometry.
Thus, even if we were able to solve the equations and specified boundary
conditions exactly, the result will not describe the flow exactly due t o the
errors in the model assumptions. We therefore define the modeling error as the
difference between the real flow and the exact solution of the mathematical
model.
Discretization Errors. Furthermore, we are seldom able to solve the governing equations exactly. Every numerical method produces approximate solutions, since various approximations have to be made to obtain an algebraic
system of equations that can be solved on computer. For example, in FV
methods one has to employ appropriate approximations for surface and volume integrals, variable values a t intermediate locations, and time integrals.
Obviously, the smaller the spatial and temporal discrete elements, the more
accurate these approximations become. Using better approximations can also
increase the accuracy; however, this is not a trivial matter as more accurate
approximations are more difficult to program, need more computing time
and storage, and may be difficult to apply to complex geometry. Usually, one
selects the approximations prior to writing a code so the spatial and temporal grid resolution are the only parameters a t user's disposal to control the
accuracy.
The same approximation may be very accurate in one part of the flow but
inaccurate elsewhere. Uniform spacing (either in space or in time) is seldom
optimal, since the flow may vary strongly locally in both space and time;
where the changes in variables are small, the errors will also be small. Thus,
with the same number of discrete elements and the same approximations, the
errors in the results may differ by an order of magnitude or more. Since the
computational effort is proportional to the number of discrete elements, their
proper distribution and size is essential for computational efficiency (the cost
of achieving the prescribed accuracy).
We define the discretization error as the difference between the exact
solution of the governing equations and the exact solution of the discrete
approximation.
Iteration Errors. The discretization process normally produces a coupled
set of non-linear algebraic equations. These are usually linearized and the linearized equations are also solved by an iterative method since direct solution
is usually too expensive.