152
3. Beyond the One-Way Wave Equation
The computational domain is 0 ::: x ::: I and 6.x = 1/80. The time derivatives
in (3.88) and (3.89) were integrated using a fourth -order Runge-Kutta method
and a small Courant number. The velocity (solid line) and the initial condition
solution is illustrated in Fig. 3.10b, which shows a very-high-resolution simulation at t = 0.5 s. Observe that the true solution is tending toward a square wave of
amplitude .J'i/2with a unit-amplitude spike at the left edge of each plateau. The
spikes are located at the nodal points in the velocity field where the flow is convergent. Away from the spike the solution is tending toward the initial value of 1/1
at the divergent node. A comparlson of the two differential-difference solutions is
shown at t = 1.0 s in Fig. 3.l0c and at t = 2.2 s in Fig. 3.l0d. In both Fig. 3.lOc
and Fig. 3.lOd the solution obtained with the nonaveraging scheme (3.88) is dorninated by a growing 26.x component. In contrast, the solution calculated with the
averaging scheme (3.89) never develops a large-amplitude 26.x component. At
the earlier time (Fig. 3.1Oe) the averaging scheme generates a solution that is reasonably accurate and far superior to the nonaveraging result. This superiority is
lost, however, by the later time (Fig. 3.lOd), at which the averaging scheme has
generated a longer-wavelength disturbance that rapidly amplifies and dominates
the solution.
Comparison 0/the Aliasing Error in Two Schemes
As suggested by the preceding example, one important difference between the
nonaveraging scheme (3.88) and the averaging method (3.89) lies in the nature of
the aliasing error generated by each approximation. Let us examine the aliasing
error produced by each formula in a more general context. Suppose that numerical
solutions are sought to the one-dimensional advection equation (3.87), and that at
some instant in time, c and interaction of the individual pair of modes:
(3.97)
Ifthe unapproximated product of c and ö on a discrete mesh, one obtains
(3.98)
Evaluating c times the nonaveraging spatial-difference operator 02x mesh gives
Cj02x iCk ak
'_2(sink26.x)e"
.(k +k 2JIJoX.
) . A
6.x
(3.99)
3. Beyond the One-Way Wave Equation
The computational domain is 0 ::: x ::: I and 6.x = 1/80. The time derivatives
in (3.88) and (3.89) were integrated using a fourth -order Runge-Kutta method
and a small Courant number. The velocity (solid line) and the initial condition
solution is illustrated in Fig. 3.10b, which shows a very-high-resolution simulation at t = 0.5 s. Observe that the true solution is tending toward a square wave of
amplitude .J'i/2with a unit-amplitude spike at the left edge of each plateau. The
spikes are located at the nodal points in the velocity field where the flow is convergent. Away from the spike the solution is tending toward the initial value of 1/1
at the divergent node. A comparlson of the two differential-difference solutions is
shown at t = 1.0 s in Fig. 3.l0c and at t = 2.2 s in Fig. 3.l0d. In both Fig. 3.lOc
and Fig. 3.lOd the solution obtained with the nonaveraging scheme (3.88) is dorninated by a growing 26.x component. In contrast, the solution calculated with the
averaging scheme (3.89) never develops a large-amplitude 26.x component. At
the earlier time (Fig. 3.1Oe) the averaging scheme generates a solution that is reasonably accurate and far superior to the nonaveraging result. This superiority is
lost, however, by the later time (Fig. 3.lOd), at which the averaging scheme has
generated a longer-wavelength disturbance that rapidly amplifies and dominates
the solution.
Comparison 0/the Aliasing Error in Two Schemes
As suggested by the preceding example, one important difference between the
nonaveraging scheme (3.88) and the averaging method (3.89) lies in the nature of
the aliasing error generated by each approximation. Let us examine the aliasing
error produced by each formula in a more general context. Suppose that numerical
solutions are sought to the one-dimensional advection equation (3.87), and that at
some instant in time, c and interaction of the individual pair of modes:
(3.97)
Ifthe unapproximated product of c and ö on a discrete mesh, one obtains
(3.98)
Evaluating c times the nonaveraging spatial-difference operator 02x mesh gives
Cj02x iCk ak
'_2(sink26.x)e"
.(k +k 2JIJoX.
) . A
6.x
(3.99)