78
4. Finite Volume Methods
the right-hand side vanishes and the leading error term is then proportional to
(Ax)'. When the grid is non-uniform, the leading error term is proportional
to the product of Ax and the grid expansion factor minus unity. In spite of
the formal first-order accuracy, the error reduction when the grid is refined is
similar to that of a second-order approximation even on non-uniform grids.
See Sect. 3.3.4 for a detailed explanation of this behavior.
4.4.3 Quadratic Upwind Interpolation ( Q U I C K )
The next logical improvement is to approximate the variable profile between
P and E by a parabola rather than a straight line. To construct a parabola we
need to use data at one more point; in accord with the nature of convection,
the third point is taken on the upstream side, i.e. W if the flow is from P to
E (i.e. u, > 0) or EE if u, < 0, see Fig. 4.2. We thus obtain:
where D, U, and UU denote the downstream, the first upstream, and the
second upstream node, respectively (E, P, and W or P, E, and EE, depending
on the flow direction). The coefficients gl and g2 can be expressed in terms
of the nodal coordinates by:
For uniform grids, the coefficients of the three nodal values involved in the
interpolation turn out to be: for the downstream point, for the first upstream node and - for the second upstream node. This scheme is somewhat
more complex than the CDS scheme: it extends the computational molecule
one more node in each direction (in 2D, the nodes EE, WW, NN and SS
are included), and, on non-orthogonal and/or non-uniform grids, the expressions for the coefficients gi are not simple. Leonard (1979) made this scheme
popular and gave it the name QUICK (Quadratic Upwind Interpolation for
Convective Kinematics).
This quadratic interpolation scheme has a third-order truncation error on
both uniform and non-uniform grids. This can be shown by eliminating the
second derivative from Eq. (4.15) using q5w, which, on a uniform Cartesian
grid with u, > 0, leads to:
The first three terms on the right-hand side represent the QUICK approximation, while the last term is the principal truncation error. When this
interpolation scheme is used in conjunction with the midpoint-rule approximation of the surface integral, the overall approximation is, however, still of
4. Finite Volume Methods
the right-hand side vanishes and the leading error term is then proportional to
(Ax)'. When the grid is non-uniform, the leading error term is proportional
to the product of Ax and the grid expansion factor minus unity. In spite of
the formal first-order accuracy, the error reduction when the grid is refined is
similar to that of a second-order approximation even on non-uniform grids.
See Sect. 3.3.4 for a detailed explanation of this behavior.
4.4.3 Quadratic Upwind Interpolation ( Q U I C K )
The next logical improvement is to approximate the variable profile between
P and E by a parabola rather than a straight line. To construct a parabola we
need to use data at one more point; in accord with the nature of convection,
the third point is taken on the upstream side, i.e. W if the flow is from P to
E (i.e. u, > 0) or EE if u, < 0, see Fig. 4.2. We thus obtain:
where D, U, and UU denote the downstream, the first upstream, and the
second upstream node, respectively (E, P, and W or P, E, and EE, depending
on the flow direction). The coefficients gl and g2 can be expressed in terms
of the nodal coordinates by:
For uniform grids, the coefficients of the three nodal values involved in the
interpolation turn out to be: for the downstream point, for the first upstream node and - for the second upstream node. This scheme is somewhat
more complex than the CDS scheme: it extends the computational molecule
one more node in each direction (in 2D, the nodes EE, WW, NN and SS
are included), and, on non-orthogonal and/or non-uniform grids, the expressions for the coefficients gi are not simple. Leonard (1979) made this scheme
popular and gave it the name QUICK (Quadratic Upwind Interpolation for
Convective Kinematics).
This quadratic interpolation scheme has a third-order truncation error on
both uniform and non-uniform grids. This can be shown by eliminating the
second derivative from Eq. (4.15) using q5w, which, on a uniform Cartesian
grid with u, > 0, leads to:
The first three terms on the right-hand side represent the QUICK approximation, while the last term is the principal truncation error. When this
interpolation scheme is used in conjunction with the midpoint-rule approximation of the surface integral, the overall approximation is, however, still of