Artificial Viscous Diffusion

Consider again the one-dimensional convective wave equation in dimensionless form as follows:

Подпись: (7.20)

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d U d U

+ = 0 —ж < x < ж.

dt dx

Подпись: (7.21)

Suppose Eq. (7.20) is discretized according to the DRP scheme with the addition of artificial selective damping terms. The discretized equations are

Подпись: u(x, 0) = 0.5e Подпись: ta2(!)2 Подпись: (7.22)

For the stated purpose, RA1 is taken to be equal to 5. This is about 15 times more damping than what was used in the previous example. Suppose the initial condition is

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Figure 7.5. Effects of excessive artificial damping on an acoustic pulse… , waveform with­out damping; , waveform with large damping R-1 = 5, t = 400.

In Chapter 2, it was demonstrated that the 7-point stencil DRP scheme can provide an almost error-free solution for this initial value problem when there is no artificial damping. Figure 7.5 shows the computed result at t = 400 by using a damping stencil with a = 0.3 n and At = 0.02. By comparing the computed waveform and the exact solution, it is clear that there is an overall reduction in the wave amplitude. However, at points farther away from the center of the pulse, the wave amplitude has actually increased, as if the pulse has diffused outward on both sides. This is a very unexpected phenomenon.

The cause of this apparent diffusion effect is subtle but can be understood by viewing the damping process in wave number space. Since the Fourier transform of a Gaussian function is also a Gaussian function, the initial pulse is also a concentrated pulse around a Ax = 0 in the wave number space. Now in time, the artificial selective damping terms gradually reduce the amplitude of the pulse in the wave number space, but not evenly. By design, there is no reduction at zero wave number. The reduction increases with wave number. Because of this, the half-width of the pulse in wave number space is reduced over time. The maximum height that occurs at a Ax = 0, nevertheless, remains the same. The waveform in the physical space is the Fourier inverse transform of the waveform in the wave number space. With a narrower pulse, as time increases, the physical waveform spreads out according to the Inverse Spreading Theorem of Fourier transform. Thus, artificial selective damping has the unintended side effect of inducing artificial viscous diffusion.

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x/Ax

(c)

Figure 7.6. Numerical solution showing the onset of instability due to excessive damping.

R-1 = 15 ………. , waveform without damping;____ , waveform with damping. (a) t = 140,

(b) t = 150, (c) t = 400.

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