Page 186 - Computational Fluid Dynamics for Engineers

References 173

a b c
Fig. 5.6. Effects of dissipation and dispersion, (a) Exact solution (b) numerical solution
distorted primarily by dissipation errors (typical of first-order methods), (c) Numerical
solution distorted primarily by dispersion errors (typical of second-order methods) [1].

of purely numerical origin and has no physical significance. The appearance of
this term in the numerical solution is called numerical dissipation. The coeffi-
2
2
cient of d u/dx , namely, cAx/2(1 — a) and those like it, which act like the
physical viscosity, are called the artificial viscosity.
Numerical dispersion is another effect also observed in numerical schemes.
It arises as a result of the odd derivative terms that appear in the truncation er-
ror. It creates a numerical behavior different from that of numerical dissipation.
Dispersion results in a distortion of the different phases of a wave which shows
up as "wiggles" in front of and behind the wave. The combined effect of dissi-
pation and dispersion is sometimes referred to as diffusion. Diffusion tends to
spread-out sharp dividing lines which may appear in the computational domain.
Figure 5.6 illustrates the effects of dissipation and dispersion on the computa-
tion of a discontinuity taken from [1]. In general, if the lowest-order term in
the truncation error contains an even derivative, the resulting solution will pre-
dominately exhibit dissipative errors. On the other hand, if the leading term is
an odd derivative, the resulting solution will predominately exhibit dispersive
errors.
In Chapters 10 and 12 we will discuss the incorporation of additional terms
to the Euler and Navier-Stokes equations in order to reduce or eliminate the
dissipation and dispersion errors and have stable numerical solutions.

References

[1] Hirsch, C : Numerical Computation of Internal and External Flows, Vol. 1, John Wiley
and Sons, N.Y., 1988.
[2] Kress, H. O.: On difference approximations of the dissipative type for hyperbolic dif-
ferential equations, Comm. Pure and Applied Mathematics 17, 335-353, 1964.
[3] Briley, W. R., McDonald, H.: Solution of the three-dimensional compressible Navier-
Stokes equations by an implicit technique, Proc. Fourth International Conference on
Numerical Methods in Fluid Dynamics, Lecture Notes in Physics, Vol. 35, Springer-
Verlag, Berlin Heidelberg, 1975.
[4] Beam, R. M. and Warming R. F.: Alternating direction implicit methods for parabolic
equations with a mixed derivative, SIAM J. of Sci. Stat. Comp. 1, 131-159, 1980.