34
2. Introduction to Numerical Methods
physically realistic solutions. This is not a numerical issue per se but models
that are not realizable may result in unphysical solutions or cause numerical
methods to diverge. We shall not deal with these issues in this book, but if
one wants to implement a model in a CFD code, one has t o be careful about
this property.
2.5.7 Accuracy
Numerical solutions of fluid flow and heat transfer problems are only approximate solutions. In addition to the errors that might be introduced in
the course of the development of the solution algorithm, in programming or
setting up the boundary conditions, numerical solutions always include three
kinds of systematic errors:
0 Modeling errors, which are defined as the difference between the actual
flow and the exact solution of the mathematical model;
0 Discretization errors, defined as the difference between the exact solution
of the conservation equations and the exact solution of the algebraic system
of equations obtained by discretizing these equations, and
Iteration errors, defined as the difference between the iterative and exact
solutions of the algebraic equations systems.
Iteration errors are often called convergence errors (which was the case in
the earlier editions of this book). However, the term convergence is used
not only in conjunction with error reduction in iterative solution methods,
but is also (quite appropriately) often associated with the convergence of
numerical solutions towards a grid-independent solution, in which case it is
closely linked t o discretization error. To avoid confusion, we shall adhere t o
the above definition of errors and, when discussing issues of convergence,
always indicate which type of convergence we are talking about.
It is important to be aware of the existence of these errors, and even more
to try t o distinguish one from another. Various errors may cancel each other,
so that sometimes a solution obtained on a coarse grid may agree better with
the experiment than a solution on a finer grid - which, by definition, should
be more accurate.
Modeling errors depend on the assumptions made in deriving the transport equations for the variables. They may be considered negligible when
laminar flows are investigated, since the Navier-Stokes equations represent
a sufficiently accurate model of the flow. However, for turbulent flows, twophase flows, combustion etc., the modeling errors may be very large - the
exact solution of the model equations may be qualitatively wrong. Modeling
errors are also introduced by simplifying the geometry of the solution domain, by simplifying boundary conditions etc. These errors are not known a
priori; they can only be evaluated by comparing solutions in which the discretization and convergence errors are negligible with accurate experimental
data or with data obtained by more accurate models ( e g data from direct
2. Introduction to Numerical Methods
physically realistic solutions. This is not a numerical issue per se but models
that are not realizable may result in unphysical solutions or cause numerical
methods to diverge. We shall not deal with these issues in this book, but if
one wants to implement a model in a CFD code, one has t o be careful about
this property.
2.5.7 Accuracy
Numerical solutions of fluid flow and heat transfer problems are only approximate solutions. In addition to the errors that might be introduced in
the course of the development of the solution algorithm, in programming or
setting up the boundary conditions, numerical solutions always include three
kinds of systematic errors:
0 Modeling errors, which are defined as the difference between the actual
flow and the exact solution of the mathematical model;
0 Discretization errors, defined as the difference between the exact solution
of the conservation equations and the exact solution of the algebraic system
of equations obtained by discretizing these equations, and
Iteration errors, defined as the difference between the iterative and exact
solutions of the algebraic equations systems.
Iteration errors are often called convergence errors (which was the case in
the earlier editions of this book). However, the term convergence is used
not only in conjunction with error reduction in iterative solution methods,
but is also (quite appropriately) often associated with the convergence of
numerical solutions towards a grid-independent solution, in which case it is
closely linked t o discretization error. To avoid confusion, we shall adhere t o
the above definition of errors and, when discussing issues of convergence,
always indicate which type of convergence we are talking about.
It is important to be aware of the existence of these errors, and even more
to try t o distinguish one from another. Various errors may cancel each other,
so that sometimes a solution obtained on a coarse grid may agree better with
the experiment than a solution on a finer grid - which, by definition, should
be more accurate.
Modeling errors depend on the assumptions made in deriving the transport equations for the variables. They may be considered negligible when
laminar flows are investigated, since the Navier-Stokes equations represent
a sufficiently accurate model of the flow. However, for turbulent flows, twophase flows, combustion etc., the modeling errors may be very large - the
exact solution of the model equations may be qualitatively wrong. Modeling
errors are also introduced by simplifying the geometry of the solution domain, by simplifying boundary conditions etc. These errors are not known a
priori; they can only be evaluated by comparing solutions in which the discretization and convergence errors are negligible with accurate experimental
data or with data obtained by more accurate models ( e g data from direct