2.3 Implementation of Numerical Solution
29
Fig. 2.4 Illustration of second-order upwind scheme
u
∂ϕ
∂ x
≈
u i
2x
(−3ϕ i + 4ϕ i+1 − ϕ i+2 ), u i < 0
(2.29b)
The interface interpolation of the second order upwind is defined as follows (taking
ϕ w for example)
ϕ w = 1.5ϕ W − 0.5ϕ W W , u W > 0
ϕ w = 1.5ϕ P − 0.5ϕ E , u W < 0
(2.30)
When the convection term adopts the second-order upwind scheme and the diffusion term adopts the central difference scheme, the obtained discrete equation has a
second-order precision truncation error.
(5) QUICK scheme
The QUICK scheme improves the truncation error of the scheme by increasing the
interpolation function on the interface.
A curvature correction rate correction method introduced by Leonard based on
piecewise linear interpolation for the quadratic upwind scheme is
ϕ e =
ϕ P + ϕ E
2
−
1
8
Curv
(2.31)
where symbol Curv represents the correction of curvature.
QUICK is the English abbreviation for “Quadratic Upwind Interpolation of
Convective Kinematics”. “Quadratic” is relative to linear interpolation (“primary”);
“upward” means that the curvature correction Curv is always composed of two points
on both sides of the curved interface and another point in the upward direction. The
QUICK scheme of the convection term has a third-order precision truncation error,
29
Fig. 2.4 Illustration of second-order upwind scheme
u
∂ϕ
∂ x
≈
u i
2x
(−3ϕ i + 4ϕ i+1 − ϕ i+2 ), u i < 0
(2.29b)
The interface interpolation of the second order upwind is defined as follows (taking
ϕ w for example)
ϕ w = 1.5ϕ W − 0.5ϕ W W , u W > 0
ϕ w = 1.5ϕ P − 0.5ϕ E , u W < 0
(2.30)
When the convection term adopts the second-order upwind scheme and the diffusion term adopts the central difference scheme, the obtained discrete equation has a
second-order precision truncation error.
(5) QUICK scheme
The QUICK scheme improves the truncation error of the scheme by increasing the
interpolation function on the interface.
A curvature correction rate correction method introduced by Leonard based on
piecewise linear interpolation for the quadratic upwind scheme is
ϕ e =
ϕ P + ϕ E
2
−
1
8
Curv
(2.31)
where symbol Curv represents the correction of curvature.
QUICK is the English abbreviation for “Quadratic Upwind Interpolation of
Convective Kinematics”. “Quadratic” is relative to linear interpolation (“primary”);
“upward” means that the curvature correction Curv is always composed of two points
on both sides of the curved interface and another point in the upward direction. The
QUICK scheme of the convection term has a third-order precision truncation error,