328
6. Semi-Lagrangian Methods
Y(x)
. .
.
. . . . . . . . . .
, , , ,
,
--.....;.a--.;. - •
tx rXp-2
Xp_1
x x p
Xp+l
Xp+2
FIGURE 6.4. Interpolation via the solution of a constant-wind-speed advection problem.
The initial condition is indicated by the solid line; the solution after translation a distance
a is indicated by the dashed line.
resolved gradients can still be minimized by approximating the solution to the
parametrized advection equation using any of the various flux-limited or fluxcorrected advection schemes discussed in Chapter 5.
If f(x) is a continuously differentiable function, the value of f at some arbitrary point x can be estimated from its value on a regularly spaced mesh by
computing a numerical solution to the constant-coefficient advection problem
ay ay
-
ar
+ -
ax
= 0
(6.55)
subject to the initial condition Y (x, 0) = f (x). The solution to this advection
problem is Y(x, r) = f(x - r) . Let x p be the x-coordinate of the grid point
nearest to xand define tx = x p - x; then
Y(x p , a) = f(x p - a) = f(x).
Figure 6.4 illustrates how the initial distribution of Y is shifted along the xcoordinate so that desired value of f(x) becomes the value of Y at grid point
x p when r = a.
If a single time step is used to integrate forward or backward over the interval
lH = o, the magnitude of the Courant number associated with this integration
will be lai Ar] . Since lai ßxl < ! by the definition of a, stable estimates of f(x)
can be obtained in a single time step using most of the wide variety of schemes
available for the numerical approximation of (6.55). Of course, there is no advantage to this approach if (6.55) is solved using an elementary scheme like the
Lax-Wendroff method, which will yield exactly the same result that would be
obtained if f (x) was interpolated from the quadratic polynomial passing through
the points f(x p +)), f(x p ) , and f(x p - )) . As discussed previously, the advantage
of this approach lies in the possibility of using positive definite or flux-limited
advection schemes to eliminate spurious negative concentrations or minimize un-
6. Semi-Lagrangian Methods
Y(x)
. .
.
. . . . . . . . . .
, , , ,
,
--.....;.a--.;. - •
tx rXp-2
Xp_1
x x p
Xp+l
Xp+2
FIGURE 6.4. Interpolation via the solution of a constant-wind-speed advection problem.
The initial condition is indicated by the solid line; the solution after translation a distance
a is indicated by the dashed line.
resolved gradients can still be minimized by approximating the solution to the
parametrized advection equation using any of the various flux-limited or fluxcorrected advection schemes discussed in Chapter 5.
If f(x) is a continuously differentiable function, the value of f at some arbitrary point x can be estimated from its value on a regularly spaced mesh by
computing a numerical solution to the constant-coefficient advection problem
ay ay
-
ar
+ -
ax
= 0
(6.55)
subject to the initial condition Y (x, 0) = f (x). The solution to this advection
problem is Y(x, r) = f(x - r) . Let x p be the x-coordinate of the grid point
nearest to xand define tx = x p - x; then
Y(x p , a) = f(x p - a) = f(x).
Figure 6.4 illustrates how the initial distribution of Y is shifted along the xcoordinate so that desired value of f(x) becomes the value of Y at grid point
x p when r = a.
If a single time step is used to integrate forward or backward over the interval
lH = o, the magnitude of the Courant number associated with this integration
will be lai Ar] . Since lai ßxl < ! by the definition of a, stable estimates of f(x)
can be obtained in a single time step using most of the wide variety of schemes
available for the numerical approximation of (6.55). Of course, there is no advantage to this approach if (6.55) is solved using an elementary scheme like the
Lax-Wendroff method, which will yield exactly the same result that would be
obtained if f (x) was interpolated from the quadratic polynomial passing through
the points f(x p +)), f(x p ) , and f(x p - )) . As discussed previously, the advantage
of this approach lies in the possibility of using positive definite or flux-limited
advection schemes to eliminate spurious negative concentrations or minimize un-