Page 184 - Computational Fluid Dynamics for Engineers

5.8 Numerical Dissipation and Dispersion: Artificial Viscosity 171

accuracy determined by the truncation and round-off errors, it is useful to dis-
cuss the numerical dissipation and dispersion errors these equations have. The
numerical dissipation arises as a result of the even derivative term that appears
in the truncation error and the numerical dispersion arises as a result of the odd
derivative term that appears in the truncation error. The numerical solutions
can be distorted by either (or both) of these errors. With the concept of artificial
viscosity, however, the stability of the numerical solutions can be improved, as
will be discussed in Chapters 10 and 12.
Consider the one-dimensional wave equation given by Eq. (5.1.1), that is
du du
= 0 c> 0 (5.1.1)
dt dx
Using a first-order forward difference in time and a first-order backward differ-
ence in space, this equation can be written as
,n+l nin
U, W
+ c- = 0 (5.8.1)
At Ax
This is a first-order accurate method with truncation error of 0(At, Ax).
We use Taylor's series expansion, see Eq. (4.3.1), and write

,n+l At + (5.8.2a)
«' + ai\ w).
(du\ n .
2
&r J. 2 (5.8.2b)
— u 4
Substituting the above relations into Eq. (5.8.1) and rearranging, we get
3
du\ n (du\ n At d uY {Atf 'cPuY (Ax)
2
2
2
dt)i C \dx) i dt ). ~2~ \dt 3 J + Ox ). ° 2
3
'd uY (Ax)
3
dx I 6 (5.8.3)
The left-hand side of this equation corresponds to Eq. (5.1.1) and the right-
hand side is the truncation error associated with Eq. (5.8.1). The importance of
terms in the truncation error can be better appreciated if we first differentiate
Eq. (5.8.3) with respect to t; multiply Eq. (5.8.3) with c after differentiating
with respect to x and subtracting the resulting equation from the equation
differentiated with respect to t, we obtain

2 U 2 u 3 3
d U _ 2 ,d i At d u + d c u +0{At)
W = c dx^ + Y W Wo~x
3
3
Ax d u 2 2d u _. A .
+ c^^-c —,+0(Ax) 3 (5.8.4)
2
dx dt
dx