Eliminating the 0
2.4 Space-Differencing
term between these two expressions, one obtains
87
Expanding the operators in the preceding yields the following 0
tridiagonal system for df/dx:
(1402xfj + Ö4x/j) = [
j+l + 3
j +
j - J .
(2.82)
When compact schemes are used to approximate partial derivatives in complex
equations in which one must compute several different spatial derivatives, such
as the multidimensional advection equation, it is simplest to solve either (2.81) or
(2.82) as aseparate tridiagonal system for each derivative. However, in very simple problems, such as the one -dimensional advection equation (2.59), the spatial
derivatives in the compact fonnulae may be replaced directly by -(l/e)B",,/Bt.
Thus, in order to analyze the phase-speed error associated with compact spatial
differencing, the fourth -order compact approximation to the advection equation
may be written
rPj+l - rPj-1
_
- 6e [(BrP) Bt j+1
+ 4 (BrP) + (BrP)
Bt j Bt j_1 ]
(2.83)
.
Substitution of a wave solution of the form (2.61) into the preceding yields the
following expression for the phase speed of the differential-difference solution:
W4p
3e
e4c = T = 2 + cos k
k
(2.84)
.
The phase speeds for the sixth-order compact scheme,
e
e6c = 3(3 +
1
+
k Sx
,
may be obtained through a similar derivation. These phase speeds are plotted as
a function of k Sx , together with the curves for second-, fourth-, and sixth-order
explicit centered differences, in Fig . 2.17. It is apparent that the compact schemes
are superior to the explicit schemes. In particular, the phase speeds associated with
the sixth-order compact differencing are almost perfect for wavelengths as short
as
Note that although the order of accuracy of a scheme detennines the rate
at which the phase-speed curves in Fig . 2.17 asymptotically approach the correct
value as
0, the order of accuracy does not reliably predict a scheme's
ability to represent the poorly resolved waves. Lele (1992) observed that a better
treatment of the shorter waves can be obtained by perturbing the coefficients in
2.4 Space-Differencing
term between these two expressions, one obtains
87
Expanding the operators in the preceding yields the following 0
tridiagonal system for df/dx:
(1402xfj + Ö4x/j) = [
j+l + 3
j +
j - J .
(2.82)
When compact schemes are used to approximate partial derivatives in complex
equations in which one must compute several different spatial derivatives, such
as the multidimensional advection equation, it is simplest to solve either (2.81) or
(2.82) as aseparate tridiagonal system for each derivative. However, in very simple problems, such as the one -dimensional advection equation (2.59), the spatial
derivatives in the compact fonnulae may be replaced directly by -(l/e)B",,/Bt.
Thus, in order to analyze the phase-speed error associated with compact spatial
differencing, the fourth -order compact approximation to the advection equation
may be written
rPj+l - rPj-1
_
- 6e [(BrP) Bt j+1
+ 4 (BrP) + (BrP)
Bt j Bt j_1 ]
(2.83)
.
Substitution of a wave solution of the form (2.61) into the preceding yields the
following expression for the phase speed of the differential-difference solution:
W4p
3e
e4c = T = 2 + cos k
k
(2.84)
.
The phase speeds for the sixth-order compact scheme,
e
e6c = 3(3 +
1
+
k Sx
,
may be obtained through a similar derivation. These phase speeds are plotted as
a function of k Sx , together with the curves for second-, fourth-, and sixth-order
explicit centered differences, in Fig . 2.17. It is apparent that the compact schemes
are superior to the explicit schemes. In particular, the phase speeds associated with
the sixth-order compact differencing are almost perfect for wavelengths as short
as
Note that although the order of accuracy of a scheme detennines the rate
at which the phase-speed curves in Fig . 2.17 asymptotically approach the correct
value as
0, the order of accuracy does not reliably predict a scheme's
ability to represent the poorly resolved waves. Lele (1992) observed that a better
treatment of the shorter waves can be obtained by perturbing the coefficients in