26
2. Introduction to Numerical Methods
Each type of method yields the same solution if the grid is very fine.
However, some methods are more suitable to some classes of problems than
others. The preference is often determined by the attitude of the developer.
We shall discuss the pros and cons of the various methods later.
2.4.3 Coordinate and Basis Vector Systems
It was mentioned in Chap. 1 that the conservation equations can be written
in many different forms, depending on the coordinate system and the basis
vectors used. For example one can select Cartesian, cylindrical, spherical,
curvilinear orthogonal or non-orthogonal coordinate systems, which may be
fixed or moving. The choice depends on the target flow, and may influence
the discretization method and grid type to be used.
One also has t o select the basis in which vectors and tensors will be defined
(fixed or variable, covariant or contravariant, etc.). Depending on this choice,
the velocity vector and stress tensor can be expressed in terms of e.g. Cartesian, covariant or contravariant, physical or non-physical coordinate-oriented
components. In this book we shall use Cartesian components exclusively for
reasons explained in Chap. 8.
2.4.4 Numerical Grid
The discrete locations at which the variables are to be calculated are defined
by the numerical grid which is essentially a discrete representation of the
geometric domain on which the problem is to be solved. It divides the solution
domain into a finite number of subdomains (elements, control volumes etc.).
Some of the options available are the following:
0 Structured (regular) grid - Regular or structured grids consist of families
of grid lines with the property that members of a single family do not cross
each other and cross each member of the other families only once. This
allows the lines of a given set to be numbered consecutively. The position of
any grid point (or control volume) within the domain is uniquely identified
by a set of two (in 2D) or three (in 3D) indices, e.g. (i, j, k).
This is the simplest grid structure, since it is logically equivalent to a Cartesian grid. Each point has four nearest neighbors in two dimensions and six
in three dimensions; one of the indices of each neighbor of point P (indices
i, j, k) differs by f 1 from the corresponding index of P. An example of a
structured 2D grid is shown in Fig. 2.1. This neighbor connectivity simplifies programming and the matrix of the algebraic equation system has
a regular structure, which can be exploited in developing a solution technique. Indeed, there is a large number of efficient solvers applicable only to
structured grids (see Chap. 5). The disadvantage of structured grids is that
they can be used only for geometrically simple solution domains. Another
disadvantage is that it may be difficult to control the distribution of the
2. Introduction to Numerical Methods
Each type of method yields the same solution if the grid is very fine.
However, some methods are more suitable to some classes of problems than
others. The preference is often determined by the attitude of the developer.
We shall discuss the pros and cons of the various methods later.
2.4.3 Coordinate and Basis Vector Systems
It was mentioned in Chap. 1 that the conservation equations can be written
in many different forms, depending on the coordinate system and the basis
vectors used. For example one can select Cartesian, cylindrical, spherical,
curvilinear orthogonal or non-orthogonal coordinate systems, which may be
fixed or moving. The choice depends on the target flow, and may influence
the discretization method and grid type to be used.
One also has t o select the basis in which vectors and tensors will be defined
(fixed or variable, covariant or contravariant, etc.). Depending on this choice,
the velocity vector and stress tensor can be expressed in terms of e.g. Cartesian, covariant or contravariant, physical or non-physical coordinate-oriented
components. In this book we shall use Cartesian components exclusively for
reasons explained in Chap. 8.
2.4.4 Numerical Grid
The discrete locations at which the variables are to be calculated are defined
by the numerical grid which is essentially a discrete representation of the
geometric domain on which the problem is to be solved. It divides the solution
domain into a finite number of subdomains (elements, control volumes etc.).
Some of the options available are the following:
0 Structured (regular) grid - Regular or structured grids consist of families
of grid lines with the property that members of a single family do not cross
each other and cross each member of the other families only once. This
allows the lines of a given set to be numbered consecutively. The position of
any grid point (or control volume) within the domain is uniquely identified
by a set of two (in 2D) or three (in 3D) indices, e.g. (i, j, k).
This is the simplest grid structure, since it is logically equivalent to a Cartesian grid. Each point has four nearest neighbors in two dimensions and six
in three dimensions; one of the indices of each neighbor of point P (indices
i, j, k) differs by f 1 from the corresponding index of P. An example of a
structured 2D grid is shown in Fig. 2.1. This neighbor connectivity simplifies programming and the matrix of the algebraic equation system has
a regular structure, which can be exploited in developing a solution technique. Indeed, there is a large number of efficient solvers applicable only to
structured grids (see Chap. 5). The disadvantage of structured grids is that
they can be used only for geometrically simple solution domains. Another
disadvantage is that it may be difficult to control the distribution of the