80
4. Finite Volume Methods
of the polynomial (4.20) by fitting it to the variable values and first derivatives
a t the two nodes on either side of the cell face. For a uniform Cartesian grid,
the following expression for 4, results:
The first term on the right-hand side of the above equation represents secondorder approximation by linear interpolation; the second term represents an
approximation of the second derivative which occurs in the leading truncation
error term for linear interpolation, see Eq. (4.15).
The problem is that the derivatives at nodes P and E are not known and
must themselves be approximated. However, even if we approximate the first
derivatives by a second-order CDS, i.e.:
the resulting approximation of the cell-face value retains the fourth-order
accuracy of the polynomial:
If we use as data the variable values on either side of the cell face and
the derivative on the upstream side, we can fit a parabola. This leads to an
approximation equivalent to the QUICK scheme described above:
The same approach can be used t o obtain an approximation of the derivative at the cell-face center; from the derivative of the polynomial (4.20) we
obtain:
Obviously, the first term on the right-hand side is the second-order CDS
approximation. The remaining terms represent a correction which increases
the accuracy.
The problem with approximations (4.24), (4.26) and (4.27) is that they
contain first derivatives a t CV centers, which are not known. Although we
can replace these by second-order approximations expressed in terms of the
nodal variable values without destroying their order of accuracy, the resulting
computational molecules will be much larger than we would like them to
be. For example, in 2D, using Simpson's rule and fourth-order polynomial
4. Finite Volume Methods
of the polynomial (4.20) by fitting it to the variable values and first derivatives
a t the two nodes on either side of the cell face. For a uniform Cartesian grid,
the following expression for 4, results:
The first term on the right-hand side of the above equation represents secondorder approximation by linear interpolation; the second term represents an
approximation of the second derivative which occurs in the leading truncation
error term for linear interpolation, see Eq. (4.15).
The problem is that the derivatives at nodes P and E are not known and
must themselves be approximated. However, even if we approximate the first
derivatives by a second-order CDS, i.e.:
the resulting approximation of the cell-face value retains the fourth-order
accuracy of the polynomial:
If we use as data the variable values on either side of the cell face and
the derivative on the upstream side, we can fit a parabola. This leads to an
approximation equivalent to the QUICK scheme described above:
The same approach can be used t o obtain an approximation of the derivative at the cell-face center; from the derivative of the polynomial (4.20) we
obtain:
Obviously, the first term on the right-hand side is the second-order CDS
approximation. The remaining terms represent a correction which increases
the accuracy.
The problem with approximations (4.24), (4.26) and (4.27) is that they
contain first derivatives a t CV centers, which are not known. Although we
can replace these by second-order approximations expressed in terms of the
nodal variable values without destroying their order of accuracy, the resulting
computational molecules will be much larger than we would like them to
be. For example, in 2D, using Simpson's rule and fourth-order polynomial