2.6 Discretization Approaches
35
simulation of turbulence, etc.). It is essential to control and estimate the convergence and discretization errors before the models of physical phenomena
(like turbulence models) can be judged.
We mentioned above that discretization approximations introduce errors
which decrease as the grid is refined, and that the order of the approximation
is a measure of accuracy. However, on a given grid, methods of the same order
may produce solution errors which differ by as much as an order of magnitude.
This is because the order only tells us the rate at which the error decreases
as the mesh spacing is reduced - it gives no information about the error on a
single grid. We shall show how discretization errors can be estimated in the
next chapter.
Errors due to iterative solution and round-off are easier to control; we
shall see how this can be done in Chap. 5 , where iterative solution methods
are introduced.
There are many solution schemes and the developer of a CFD code may
have a difficult time deciding which one to adopt. The ultimate goal is to
obtain desired accuracy with least effort, or the maximum accuracy with the
available resources. Each time we describe a particular scheme we shall point
out its advantages or disadvantages with respect to these criteria.
2.6 Discretization Approaches
2.6.1 Finite Difference Method
This is the oldest method for numerical solution of PDE's, believed to have
been introduced by Euler in the 18th century. It is also the easiest method
to use for simple geometries.
The starting point is the conservation equation in differential form. The
solution domain is covered by a grid. At each grid point, the differential equation is approximated by replacing the partial derivatives by approximations
in terms of the nodal values of the functions. The result is one algebraic equation per grid node, in which the variable value a t that and a certain number
of neighbor nodes appear as unknowns.
In principle, the FD method can be applied to any grid type. However, in
all applications of the FD method known to the authors, it has been applied
to structured grids. The grid lines serve as local coordinate lines.
Taylor series expansion or polynomial fitting is used to obtain approximations to the first and second derivatives of the variables with respect to the
coordinates. When necessary, these methods are also used to obtain variable
values a t locations other than grid nodes (interpolation). The most widely
used methods of approximating derivatives by finite differences are described
in Chap. 3.
On structured grids, the FD method is very simple and effective. It is
especially easy to obtain higher-order schemes on regular grids; some will be
35
simulation of turbulence, etc.). It is essential to control and estimate the convergence and discretization errors before the models of physical phenomena
(like turbulence models) can be judged.
We mentioned above that discretization approximations introduce errors
which decrease as the grid is refined, and that the order of the approximation
is a measure of accuracy. However, on a given grid, methods of the same order
may produce solution errors which differ by as much as an order of magnitude.
This is because the order only tells us the rate at which the error decreases
as the mesh spacing is reduced - it gives no information about the error on a
single grid. We shall show how discretization errors can be estimated in the
next chapter.
Errors due to iterative solution and round-off are easier to control; we
shall see how this can be done in Chap. 5 , where iterative solution methods
are introduced.
There are many solution schemes and the developer of a CFD code may
have a difficult time deciding which one to adopt. The ultimate goal is to
obtain desired accuracy with least effort, or the maximum accuracy with the
available resources. Each time we describe a particular scheme we shall point
out its advantages or disadvantages with respect to these criteria.
2.6 Discretization Approaches
2.6.1 Finite Difference Method
This is the oldest method for numerical solution of PDE's, believed to have
been introduced by Euler in the 18th century. It is also the easiest method
to use for simple geometries.
The starting point is the conservation equation in differential form. The
solution domain is covered by a grid. At each grid point, the differential equation is approximated by replacing the partial derivatives by approximations
in terms of the nodal values of the functions. The result is one algebraic equation per grid node, in which the variable value a t that and a certain number
of neighbor nodes appear as unknowns.
In principle, the FD method can be applied to any grid type. However, in
all applications of the FD method known to the authors, it has been applied
to structured grids. The grid lines serve as local coordinate lines.
Taylor series expansion or polynomial fitting is used to obtain approximations to the first and second derivatives of the variables with respect to the
coordinates. When necessary, these methods are also used to obtain variable
values a t locations other than grid nodes (interpolation). The most widely
used methods of approximating derivatives by finite differences are described
in Chap. 3.
On structured grids, the FD method is very simple and effective. It is
especially easy to obtain higher-order schemes on regular grids; some will be