Problems
169
8. Derive expressions for the boundaries for the regions of useful stability for
the leapfrog-backward scheme (3.78) and the leapfrog-trapezoidal method
(3.79) shown in Fig. 3.7a.
9. The following approximation to the advection-diffusion equation (3.72) is
unstable:
OZtrPj + cozxrPj = Mo;rPj.
Modify the right side of the above equation to stabilize the method, at least
for sufficiently small l:i.t, but do not make the scheme implicit. Prove that
your modified scheme is indeed stable for sufficiently small values of I:i.t .
It is not necessary to work out the exact range of l:i.t over which the scheme
is stable .
10. The analysis of the frozen-coefficient problem does not always correctly
indicate the behavior of solutions to partial differential equations with variable coefficients. Consider the initial value problem
at 81/! . 8 (.
- I 8x
81/!)
1/!(x,O) = fex) on the interval -00 < x < 00.
=0,
(3.128)
(a) Show that the tz-norm of the solution to this problem does not grow
with time.
(b) Freeze the coefficients at x = 0 and show that the resulting problem is
iIl-posed because its solution does not depend continuously on the initial
data. (Hint : consider
and show that 1I1/!1 (x, 0) -1/!z (x, 0) 11 is bounded, while 1I1/!1 (x, t) -1/!z (x, t) 11
can be arbitrarily large for any finite t .)
Since stable numerical solutions cannot be obtained for ill-posed problems,
the stability of a numerical approximation to (3.128) cannot be determined
by examining the stability of the family of all frozen-coefficient problems.
11. Suppose that (3.105) and (3.106) are applied to model tracer advection in
a closed reetangular domain with no velocity normal to the boundaries and
that the boundaries are located at the edges (as opposed to the centers) of
the outermost grid cells. Let the differential-difference equations generated
by each scheme be expressed as a linear system of the form (3.104). Write
down the coefficient matrix A for each scheme, and show that the matrix
associated with (3.106) is skew-symmetric, whereas that associated with
(3.105) is not.
12. *The linearized one-dimensional Rossby adjustment problem for an atmosphere with no mean wind is govemed by the equations
169
8. Derive expressions for the boundaries for the regions of useful stability for
the leapfrog-backward scheme (3.78) and the leapfrog-trapezoidal method
(3.79) shown in Fig. 3.7a.
9. The following approximation to the advection-diffusion equation (3.72) is
unstable:
OZtrPj + cozxrPj = Mo;rPj.
Modify the right side of the above equation to stabilize the method, at least
for sufficiently small l:i.t, but do not make the scheme implicit. Prove that
your modified scheme is indeed stable for sufficiently small values of I:i.t .
It is not necessary to work out the exact range of l:i.t over which the scheme
is stable .
10. The analysis of the frozen-coefficient problem does not always correctly
indicate the behavior of solutions to partial differential equations with variable coefficients. Consider the initial value problem
at 81/! . 8 (.
- I 8x
81/!)
1/!(x,O) = fex) on the interval -00 < x < 00.
=0,
(3.128)
(a) Show that the tz-norm of the solution to this problem does not grow
with time.
(b) Freeze the coefficients at x = 0 and show that the resulting problem is
iIl-posed because its solution does not depend continuously on the initial
data. (Hint : consider
and show that 1I1/!1 (x, 0) -1/!z (x, 0) 11 is bounded, while 1I1/!1 (x, t) -1/!z (x, t) 11
can be arbitrarily large for any finite t .)
Since stable numerical solutions cannot be obtained for ill-posed problems,
the stability of a numerical approximation to (3.128) cannot be determined
by examining the stability of the family of all frozen-coefficient problems.
11. Suppose that (3.105) and (3.106) are applied to model tracer advection in
a closed reetangular domain with no velocity normal to the boundaries and
that the boundaries are located at the edges (as opposed to the centers) of
the outermost grid cells. Let the differential-difference equations generated
by each scheme be expressed as a linear system of the form (3.104). Write
down the coefficient matrix A for each scheme, and show that the matrix
associated with (3.106) is skew-symmetric, whereas that associated with
(3.105) is not.
12. *The linearized one-dimensional Rossby adjustment problem for an atmosphere with no mean wind is govemed by the equations