Problems
or flux form. Let (X), X2) = (x , y) and (U), U2) = (u, v). The advective form
297
can be derived from first principles using the definition of the total derivative in
the transformed coordinates. The flux form
a1/f
- tn
I a _
I a _
+ - - (lu,lr) + - - (lv,lr) = 0
1 ax
'I'
1 ay
'I'
,
where
1 = ax ay _ ax ay
ax ay ay ar:
can also be derived from first principles using the expression for the divergence
in transformed coordinates (5.61). The flux form implies conservation of 1/f (provided that coordinate transformation is time-independent) and is ready for direct
approximation by a numerical conservation law. The numerical fluxes can be limited or corrected as discussed previously to preserve monotonicity.
The proper formulation of a numerical approximation of the advective form is
more subtle. As discussed previously, it is important to create a finite-difference
approximation to the advective form that is algebraically equivalent to the flux
form. This is achieved by the flux-limiter algorithm presented in Table 5.3, provided that the velocities satisfy the incompressible continuity equation on a staggered mesh,
I
y-. [(Ox(Ji.jÜi.j) + Oy(li'/Ü i.j)] = o.
I .)
The velocities are staggered such that Üi+!.j is located !1x/2 to the "east" and
Vi.i+! is !1y/2 to the "north" ofthe grid point where 4'i.j is defined. In the absence
of a flux limiter, the last equation in Table 5.3 is a second-order Lax-Wendroff
approximation to
a1/f + .!- (Ül a1/t + vl a1/f) = 0
at 1
ax
ay
,
where the common factor of 1 is not canceled out of the numerator and denominator because it is evaluated at different locations on the numerical mesh. The
evaluation of 1 at these slightly different grid points is required to make the finitedifference method in advective form algebraically equivalent to a scheme in flux
form.
Problems
1. Consider two sets of equations that might be supposed to govem onedimensional shallow-water flow:
or flux form. Let (X), X2) = (x , y) and (U), U2) = (u, v). The advective form
297
can be derived from first principles using the definition of the total derivative in
the transformed coordinates. The flux form
a1/f
- tn
I a _
I a _
+ - - (lu,lr) + - - (lv,lr) = 0
1 ax
'I'
1 ay
'I'
,
where
1 = ax ay _ ax ay
ax ay ay ar:
can also be derived from first principles using the expression for the divergence
in transformed coordinates (5.61). The flux form implies conservation of 1/f (provided that coordinate transformation is time-independent) and is ready for direct
approximation by a numerical conservation law. The numerical fluxes can be limited or corrected as discussed previously to preserve monotonicity.
The proper formulation of a numerical approximation of the advective form is
more subtle. As discussed previously, it is important to create a finite-difference
approximation to the advective form that is algebraically equivalent to the flux
form. This is achieved by the flux-limiter algorithm presented in Table 5.3, provided that the velocities satisfy the incompressible continuity equation on a staggered mesh,
I
y-. [(Ox(Ji.jÜi.j) + Oy(li'/Ü i.j)] = o.
I .)
The velocities are staggered such that Üi+!.j is located !1x/2 to the "east" and
Vi.i+! is !1y/2 to the "north" ofthe grid point where 4'i.j is defined. In the absence
of a flux limiter, the last equation in Table 5.3 is a second-order Lax-Wendroff
approximation to
a1/f + .!- (Ül a1/t + vl a1/f) = 0
at 1
ax
ay
,
where the common factor of 1 is not canceled out of the numerator and denominator because it is evaluated at different locations on the numerical mesh. The
evaluation of 1 at these slightly different grid points is required to make the finitedifference method in advective form algebraically equivalent to a scheme in flux
form.
Problems
1. Consider two sets of equations that might be supposed to govem onedimensional shallow-water flow: