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4. Finite Volume Methods
the cell-face values are approximated in terms of the nodal (CV center)
values.
The simplest approximation to the integral is the midpoint rule: the integral is approximated as a product of the integrand a t the cell-face center
(which is itself an approximation to the mean value over the surface) and the
cell-face area:
This approximation of the integral - provided the value of f a t location 'e'
is known - is of second-order accuracy.
Since the value of f is not available at the cell face center 'e', it has to be
obtained by interpolation. In order to preserve the second-order accuracy of
the midpoint rule approximation of the surface integral, the value of fe has
to be computed with at least second-order accuracy. We shall present some
widely used approximations in Sect. 4.4.
Another second-order approximation of the surface integral in 2D is the
trapezoid rule, which leads to:
In this case we need t o evaluate the flux at the CV corners.
For higher-order approximation of the surface integrals, the flux must
be evaluated a t more than two locations. A fourth-order approximation is
Simpson's rule, which estimates the integral over Se as:
Here the values of f are needed a t three locations: the cell face center 'e' and
the two corners, 'ne' and 'se'. In order to retain the fourth-order accuracy
these values have to be obtained by interpolation of the nodal values a t
least as accurate as Simpson's rule. Cubic polynomials are suitable, as shown
below.
In 3D, the midpoint rule is again the simplest second-order approximation.
Higher-order approximations, which require the integrand at locations other
than cell face center (e.g. corners and centers of edges) are possible, but they
are more difficult to implement. One possibility is mentioned in the following
section.
If the variation of f is assumed to have some particular simple shape (e.g.
an interpolation polynomial), the integration is easy. The accuracy of the
approximation then depends on the order of shape functions.
4. Finite Volume Methods
the cell-face values are approximated in terms of the nodal (CV center)
values.
The simplest approximation to the integral is the midpoint rule: the integral is approximated as a product of the integrand a t the cell-face center
(which is itself an approximation to the mean value over the surface) and the
cell-face area:
This approximation of the integral - provided the value of f a t location 'e'
is known - is of second-order accuracy.
Since the value of f is not available at the cell face center 'e', it has to be
obtained by interpolation. In order to preserve the second-order accuracy of
the midpoint rule approximation of the surface integral, the value of fe has
to be computed with at least second-order accuracy. We shall present some
widely used approximations in Sect. 4.4.
Another second-order approximation of the surface integral in 2D is the
trapezoid rule, which leads to:
In this case we need t o evaluate the flux at the CV corners.
For higher-order approximation of the surface integrals, the flux must
be evaluated a t more than two locations. A fourth-order approximation is
Simpson's rule, which estimates the integral over Se as:
Here the values of f are needed a t three locations: the cell face center 'e' and
the two corners, 'ne' and 'se'. In order to retain the fourth-order accuracy
these values have to be obtained by interpolation of the nodal values a t
least as accurate as Simpson's rule. Cubic polynomials are suitable, as shown
below.
In 3D, the midpoint rule is again the simplest second-order approximation.
Higher-order approximations, which require the integrand at locations other
than cell face center (e.g. corners and centers of edges) are possible, but they
are more difficult to implement. One possibility is mentioned in the following
section.
If the variation of f is assumed to have some particular simple shape (e.g.
an interpolation polynomial), the integration is easy. The accuracy of the
approximation then depends on the order of shape functions.