104
2. BasicFinite-Difference Methods
13. Determine the order of accuracy and the stability properties of the "slantderivative" approximation to the constant-wind-speed advection equation
A.'!+I _ A.'!
(A.n+l _ A.'!+1 r/J'! _ A.'!)
'l'J
= O.
'l'J
'l'J + 'l'J
'l'J-1 + J+I
ßt
2
Dox
Dox
14. Show that the upwind method of Warming and Beam (2.109) is secondorder accurate with truncation error 0 (Dot 2 + Dot Sx + DoX 2), and that it is
stable for 0 S JL S 2.
15. Suppose that the advection equation is approximated by a second-order
Runge-Kutta time difference and a centered second-order spatial difference. Show that the auxiliary condition O(Dot) S 0 [(DoX)4/3] is a necessary condition for this scheme to converge to the true solution in the limit
Dot, Dox
O.
16. Suppose that the time derivative in the differential-difference equation (2.60)
is approximated using a fourth -order Runge-Kutta Scheme. Determine the
maximum value of cDot / Dox for which this scheme will be stable using the
stability criteria for the oscillation equation given in Table 2.2. Explain how
this value can exceed unity without violating the CFL condition.
17. Derive the modified equation that is approximated through order three by
the leapfrog-time centered-space scheme (2.91). Compare this with the modified equation for the Lax-Wendroff scheme
where JL = cDot / Dox . Discuss whether the behavior of these two schemes,
as illustrated in Fig. 2.19, is consistent with the leading-order error terms in
each scheme 's modified equation.
18. Derive an expression for the upstream approximation to the constant-windspeed advection equation that remains upstream independent of the sign of
the velocity field. Express the upstream spatial derivative as the combination of a centered-space derivative and a diffusive smoother in a manner
similar to that in (2.78).
19. *Evaluate the performance of several numerical schemes for approximating
the two-component system of ordinary differential equations
du
-=fv,
dt
dv
- =
subject to the initial conditions u(O) = 1, v(O) = O. Set f = tt .
2. BasicFinite-Difference Methods
13. Determine the order of accuracy and the stability properties of the "slantderivative" approximation to the constant-wind-speed advection equation
A.'!+I _ A.'!
(A.n+l _ A.'!+1 r/J'! _ A.'!)
'l'J
= O.
'l'J
'l'J + 'l'J
'l'J-1 + J+I
ßt
2
Dox
Dox
14. Show that the upwind method of Warming and Beam (2.109) is secondorder accurate with truncation error 0 (Dot 2 + Dot Sx + DoX 2), and that it is
stable for 0 S JL S 2.
15. Suppose that the advection equation is approximated by a second-order
Runge-Kutta time difference and a centered second-order spatial difference. Show that the auxiliary condition O(Dot) S 0 [(DoX)4/3] is a necessary condition for this scheme to converge to the true solution in the limit
Dot, Dox
O.
16. Suppose that the time derivative in the differential-difference equation (2.60)
is approximated using a fourth -order Runge-Kutta Scheme. Determine the
maximum value of cDot / Dox for which this scheme will be stable using the
stability criteria for the oscillation equation given in Table 2.2. Explain how
this value can exceed unity without violating the CFL condition.
17. Derive the modified equation that is approximated through order three by
the leapfrog-time centered-space scheme (2.91). Compare this with the modified equation for the Lax-Wendroff scheme
where JL = cDot / Dox . Discuss whether the behavior of these two schemes,
as illustrated in Fig. 2.19, is consistent with the leading-order error terms in
each scheme 's modified equation.
18. Derive an expression for the upstream approximation to the constant-windspeed advection equation that remains upstream independent of the sign of
the velocity field. Express the upstream spatial derivative as the combination of a centered-space derivative and a diffusive smoother in a manner
similar to that in (2.78).
19. *Evaluate the performance of several numerical schemes for approximating
the two-component system of ordinary differential equations
du
-=fv,
dt
dv
- =
subject to the initial conditions u(O) = 1, v(O) = O. Set f = tt .