36
2. Introduction to Numerical Methods
mentioned in Chap. 3. The disadvantage of FD methods is that the conservation is not enforced unless special care is taken. Also, the restriction to
simple geometries is a significant disadvantage in complex flows.
2.6.2 Finite Volume Method
The FV method uses the integral form of the conservation equations as its
starting point. The solution domain is subdivided into a finite number of
contiguous control volumes (CVs), and the conservation equations are applied
to each CV. At the centroid of each CV lies a computational node at which the
variable values are to be calculated. Interpolation is used to express variable
values at the CV surface in terms of the nodal (CV-center) values. Surface
and volume integrals are approximated using suitable quadrature formulae.
As a result, one obtains an algebraic equation for each CV, in which a number
of neighbor nodal values appear.
The FV method can accommodate any type of grid, so it is suitable for
complex geometries. The grid defines only the control volume boundaries
and need not be related to a coordinate system. The method is conservative
by construction, so long as surface integrals (which represent convective and
diffusive fluxes) are the same for the CVs sharing the boundary.
The FV approach is perhaps the simplest to understand and to program.
All terms that need be approximated have physical meaning which is why it
is popular with engineers.
The disadvantage of FV methods compared to FD schemes is that methods of order higher than second are more difficult to develop in 3D. This is
due to the fact that the FV approach requires three levels of approximation:
interpolation, differentiation, and integration. We shall give a detailed description of the FV method in Chap. 4; it is the most used method in this
book.
2.6.3 Finite Element Method
The FE method is similar to the FV method in many ways. The domain is
broken into a set of discrete volumes or finite elements that are generally
unstructured; in 2D, they are usually triangles or quadrilaterals, while in 3D
tetrahedra or hexahedra are most often used. The distinguishing feature of
FE methods is that the equations are multiplied by a weight function before
they are integrated over the entire domain. In the simplest FE methods, the
solution is approximated by a linear shape function within each element in
a way that guarantees continuity of the solution across element boundaries.
Such a function can be constructed from its values at the corners of the
elements. The weight function is usually of the same form.
This approximation is then substituted into the weighted integral of the
conservation law and the equations to be solved are derived by requiring the
2. Introduction to Numerical Methods
mentioned in Chap. 3. The disadvantage of FD methods is that the conservation is not enforced unless special care is taken. Also, the restriction to
simple geometries is a significant disadvantage in complex flows.
2.6.2 Finite Volume Method
The FV method uses the integral form of the conservation equations as its
starting point. The solution domain is subdivided into a finite number of
contiguous control volumes (CVs), and the conservation equations are applied
to each CV. At the centroid of each CV lies a computational node at which the
variable values are to be calculated. Interpolation is used to express variable
values at the CV surface in terms of the nodal (CV-center) values. Surface
and volume integrals are approximated using suitable quadrature formulae.
As a result, one obtains an algebraic equation for each CV, in which a number
of neighbor nodal values appear.
The FV method can accommodate any type of grid, so it is suitable for
complex geometries. The grid defines only the control volume boundaries
and need not be related to a coordinate system. The method is conservative
by construction, so long as surface integrals (which represent convective and
diffusive fluxes) are the same for the CVs sharing the boundary.
The FV approach is perhaps the simplest to understand and to program.
All terms that need be approximated have physical meaning which is why it
is popular with engineers.
The disadvantage of FV methods compared to FD schemes is that methods of order higher than second are more difficult to develop in 3D. This is
due to the fact that the FV approach requires three levels of approximation:
interpolation, differentiation, and integration. We shall give a detailed description of the FV method in Chap. 4; it is the most used method in this
book.
2.6.3 Finite Element Method
The FE method is similar to the FV method in many ways. The domain is
broken into a set of discrete volumes or finite elements that are generally
unstructured; in 2D, they are usually triangles or quadrilaterals, while in 3D
tetrahedra or hexahedra are most often used. The distinguishing feature of
FE methods is that the equations are multiplied by a weight function before
they are integrated over the entire domain. In the simplest FE methods, the
solution is approximated by a linear shape function within each element in
a way that guarantees continuity of the solution across element boundaries.
Such a function can be constructed from its values at the corners of the
elements. The weight function is usually of the same form.
This approximation is then substituted into the weighted integral of the
conservation law and the equations to be solved are derived by requiring the