3.6 Nonlinear Instability
163
also develop in the conservative-form solution, but these oscillations do not continue to amplifyf The flux form (3.118) yields a solution (not shown) to this test
problem that looks qualitatively similar to the conservative-form solution shown
in Fig. 3.Bb, although the spurious oscillations in the flux-form result are actually somewhat weaker. It is perhaps surprising that the short-wavelength oscillations are smaller in the flux-form solution than in the conservative-form solution
and that the flux-form solution does not show a tendency toward instability. In
fact, practical experience suggests that the flux-form difference (3.118) is not particularly susceptible to nonlinear instability. Fomberg (1973) has, nevertheless,
demonstrated that both the advective and flux forms are unstable (and that the
conservative form is stable) when the discretized initial condition has the special
form .. . , 0, -I, 1,0, -1, 1,0, ....
The instabilities that develop in the preceding solutions to Burgers 's equation
appear to be associated with the formation of the shock. The development of a
shock is not, however, aprerequisite for the onset of nonlinear instability, and
such instabilities may occur in numerical simulations of very smooth flow, One
example in which nonlinear instability develops in a smooth flow is provided by
the viscous Burgers 's equation
81/1
81/1
8
2
1/1
at + 1/1 8x = v 8x2 '
(3.121)
where v is a coefficient of viscosity. The true solution to the viscous Burgers 's
equation never develops a shock, but the advective-form differential difference
approximation to (3.121) becomes unstable for sufficiently small values of v.
3.6.2 The Barotropic Vorticity Equation
A second example involving the development of nonlinear instability in very
smooth flow is provided by the equation goveming the vorticity in a two-dimensional incompressible homogeneous fluid,
(3.122)
Here u is the two-dimensional velocity vector describing the flow in the x-y plane
and
u and
is the vorticity component along the z-axis. Since the flow is nondivergent,
may be expressed in terms of a stream function 1/1 such that
u = k x V1/I,
8Even though they do not lead 10instability,the short-wavelength oscillations in the conservativeform solution to Burgers's equation are nonphysical and are not present in the correct generalized
solution to Burgers's equation, which satisfies the Rankine-Hugoniot condition (5.10) at the shock
and is smooth away from the shock. After the fonnation of the shock the correct generalizedsolution
ceases to conserve 114>11 2 , so it can no longer be well-approximated by the numerical solutionobtained
using the conservative-fonn difference. In order to obtain good numerical approximationsto discontinuous solutions to Burgers's equation it is necessary to use the methods discussed in Chapter 5.
163
also develop in the conservative-form solution, but these oscillations do not continue to amplifyf The flux form (3.118) yields a solution (not shown) to this test
problem that looks qualitatively similar to the conservative-form solution shown
in Fig. 3.Bb, although the spurious oscillations in the flux-form result are actually somewhat weaker. It is perhaps surprising that the short-wavelength oscillations are smaller in the flux-form solution than in the conservative-form solution
and that the flux-form solution does not show a tendency toward instability. In
fact, practical experience suggests that the flux-form difference (3.118) is not particularly susceptible to nonlinear instability. Fomberg (1973) has, nevertheless,
demonstrated that both the advective and flux forms are unstable (and that the
conservative form is stable) when the discretized initial condition has the special
form .. . , 0, -I, 1,0, -1, 1,0, ....
The instabilities that develop in the preceding solutions to Burgers 's equation
appear to be associated with the formation of the shock. The development of a
shock is not, however, aprerequisite for the onset of nonlinear instability, and
such instabilities may occur in numerical simulations of very smooth flow, One
example in which nonlinear instability develops in a smooth flow is provided by
the viscous Burgers 's equation
81/1
81/1
8
2
1/1
at + 1/1 8x = v 8x2 '
(3.121)
where v is a coefficient of viscosity. The true solution to the viscous Burgers 's
equation never develops a shock, but the advective-form differential difference
approximation to (3.121) becomes unstable for sufficiently small values of v.
3.6.2 The Barotropic Vorticity Equation
A second example involving the development of nonlinear instability in very
smooth flow is provided by the equation goveming the vorticity in a two-dimensional incompressible homogeneous fluid,
(3.122)
Here u is the two-dimensional velocity vector describing the flow in the x-y plane
and
u and
is the vorticity component along the z-axis. Since the flow is nondivergent,
may be expressed in terms of a stream function 1/1 such that
u = k x V1/I,
8Even though they do not lead 10instability,the short-wavelength oscillations in the conservativeform solution to Burgers's equation are nonphysical and are not present in the correct generalized
solution to Burgers's equation, which satisfies the Rankine-Hugoniot condition (5.10) at the shock
and is smooth away from the shock. After the fonnation of the shock the correct generalizedsolution
ceases to conserve 114>11 2 , so it can no longer be well-approximated by the numerical solutionobtained
using the conservative-fonn difference. In order to obtain good numerical approximationsto discontinuous solutions to Burgers's equation it is necessary to use the methods discussed in Chapter 5.