Numerical tests were made for a model of the Navier-Stokes equations using a second-order accurate implicit scheme which guarantees diagonal dominance. The results suggest that the failure of implicit methods using large marching steps may not always be attributed to the lack of diagonal dominance in the coefficient matrix. In some cases the failure may be caused by a nonlinear instability associated with the solution method

Hirsh, R. S.

Rudy, D. H.

English

NASA Technical Reports Server

General Disclaimer 
One or more of the Following Statements may affect this Document 
 
 This document has been reproduced from the best copy furnished by the 
organizational source. It is being released in the interest of making available as 
much information as possible. 
 
 This document may contain data, which exceeds the sheet parameters. It was 
furnished in this condition by the organizational source and is the best copy 
available. 
 
 This document may contain tone-on-tone or color graphs, charts and/or pictures, 
which have been reproduced in black and white. 
 
 This document is paginated as submitted by the original source. 
 
 Portions of this document are not fully legible due to the historical nature of some 
of the material. However, it is the best reproduction available from the original 
submission. 
 
 
 
 
 
 
 
Produced by the NASA Center for Aerospace Information (CASI) 
https://ntrs.nasa.gov/search.jsp?R=19760019342 2020-03-22T13:51:19+00:00Z
i
` t
	
J
1
NASA TECHNICAL
	
NASA TM X- 73905
MEMORANDUM
Ln0
rnMn
K
(NASA-TM-X-73905)
	 COtMENTS Ch T HE RCIE CF
DIAGONAL DCMINANCE IN IMPLICIT CIFFERENCE
METHODS (NASA)
	 1P F EC $3.50
	 CSCI 20>r
v+
QZ
N76-26430
Onclas
G3/34 44543
COMMENTS ON THE ROLE OF DIAGONAL DOMINANCE
IN IMPLICIT DIFFERENCE METHODS
By
David H. Rudy
and
Richard S. Hirsh
NASA Langley Research Center
May 1976
This informal documentation medium is used to provide accelerated or
special release of technical information to selected users. The contents
may not meet NASA formal editing and publication standards, may be re-
vised. or may be Incorporated in another publication.
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
LANGLEY RESEARCH CENTER,. HAMPTON, VIRGINIA 23665 	 JUL
NkSA sr,
INPUia^, Unt
1. Report No. 2. Govarnmert Accession No. 3. Recipient's Catalog No.
NASA TM X-73905
4. Title and Subtitle 5 Report ate
COMMENTS ON THE ROLE OF DIAGONAL DOMINANCE IN ILLICIT May 1976
6. Performing Organization CodeDIFFERENCE METHODS
7. Author(s) S. Performing Organization Report No.
Rudy, David H.; and Hirsh, Richard S.
10. Work that No.
505-06-11-03
9. Performing Organization Name and Address
NASA Langley Research Center
Hampton, VA	 23665
11. contract or Grant No.
13. Type of Report and Period Covered
NASA Technical Memorandum
12. Sponsoring Agency Name and Address
National Aeronautics and Space Administration
Washington, DC	 20546
14. sponsoring Agency Code
15. Supplementary Notes
Final release of information to be usri as a reference for a formal publication.
16. Abstract
Numerical tests were made for a model of the Navier-Stokes equations using a second-
order accurate implicit scheme which guarantees diagonal dominance. 	 The results of these
tests suggest that the failure of implicit methods us;.-.g large marching steps may not always
be attributed to the lack of diagonal dominance in the coefficient matrix. 	 In some cases
the failure may be caused by a nonlinear instability associated with the solution method.
17. Key Words (Suggested by Author(s))
	 18. Distribution Statement
Numerical analysis
Fluid mechanics	 Unclassified
Finite-difference methods
	 Unlimited
19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No. of Pages 22. Price'
Unclassi fied Unclassified 16 x3.25
For sale by theNational Technical Information Service, Springfield, Virginia 22161
COMMENTS ON THE ROLE OF DIAGONAL DOMINANCE
IN IMPLICIT DIFFERENCE METHODS
By David H. Rudy and Richard S. Hirsh
Langley Research Center
1►
SUMMARY
Numerical tests were made for a model of the Navier-Stokes equations
using a second-order accurate implicit scheme which guarantees diagonal
dominance. The results of these tests suggest that the failure of implicit
methods using large marching steps may not always be attributed to the lack of
diagonal dominance in the coefficient matrix. In some cases the failure may
be caused by a nonlinear instability associated with the solution method.
INTRODUCTION
Implicit finite difference methods in general offer the mayor advantage of
unconditional stability (with respect to the step size in the marching direc-
tion) over explicit methods. This advantage is offset somewhat by the need to
solve a system of simultaneous algebraic equations during each marching step.
However, when central differences are used, the coefficient matrix becomes
tridiagonal in form and can be inverted much more rapidly than a full matrix.
Consequently, to gain a computational time advantage, implicit marching steps
need only to be a factor of four or five times larger than the maximum step
a
	 size allowed in an explicit method. In many cases, though, large marching
steps cause the coefficient matrix to lose diagonal dominance, a sufficient,
but not necessary, condition for convergence of the matrix inversion. Without
1
2diagonal dominance, roundoff error can accumulate in the matrix inversion and
destroy the solution. For Burgers' equation (a model of the Navier-Stokes
equations), Hirsh and Rudy (Ref. 1) found that diagonal dominance could be
maintained for any size marching steps if the cell Reynolds number was 2 or
less. In the present paper results are given for numerical tests made with
the second-order accurate scheme of Khosla-Rubin (Ref. 2) which maintains
diagonal dominance for all cell Reynolds Lumbers and marching step sizes.
The results of these tests suggest that the failure of implicit methods using
large marching steps may not always be caused by the lack of diagonal dominance.
SYMBOLS
a	 =ui- U
A,B,C,D	 matrix coefficients in eqs. (3) and (6)
C	 Courant number, aAt
Ax
D'	 _ (ui+1 - 2Ui + ui-1)/(2Ax)
R 	 cell Reynolds number, +a+Ax/v
U	 velocity
U	 wave speed
Ax	 spatial coordinate mesh spacing
At	 marching direction step size
E	 convergence criterion
V	 viscosity
Subscripts
i	 index denoting grid point spatial location 	 a
max	 maximum value
x,t	 derivative with respect to x-direction, time
3
Superscripts
￿-	 n	 index denoting time level
DESCRIPTION OF NUMERICAL TESTS
The numerical tests of the Khosla-Rubin scheme were first made using
the nonconservative form of Burgers' equation
ut + aux = vu	 (1)
xx
(where a can be a function of u, t, and x) for several cases presented in
Ref. 1 where diagonal dominance was lost. The difference form of eq. (1)
becomes, using the Khosla -Rubin scheme ( ref. 2),
n+1	 n	 n+l	 n+1	 n	 n	 n
ui
 - u  
+ 
(a ; - Jail)	 (ui+l - ui ) (ui+l - 2ui + ui-1)
At	 2	 Ax	 2Ax
(a. + ja.') (un+l - un+1 )	 (un - 2u + u 	 )
+ i	 i	 i	 i-1 + i+l	 i	 i-1
	
2	 Ax	 2Ax
V (ui+l - 2ui+l + ui+l)	 (2)(Ax)2
where ai
 = ui - U and U is the steady-state wave speed. Thus,
Aiui+l + Biui	 i+l+1 + Cui - Di	(3)
where
E4
-v	 _ (ai + Jai l)	 -y	 (ai	 Jail)
Ai - 
(Ax)2
	
2Ax	 C  = (Ax)2 
+	 2Ax
M 
1 
+ 2v	
- ( ai - Jai l) + (ai
 + Jail)
Bi At 
(Ax)2
	 2Ax	 2Ax
n	 n	 n	 n
ui	 (u i
+1 - 2u  + ui-1)
D; At 
Jail
2Ax
n
= t ui _ IailDi
The boundary conditions were taken to be:
u(x,t) = 1.0, at x = -5
u(x,t) = 0, at x = +5
Fifty-one equally spaced mesh points were used with Ax = 0.2. The wave speed
U was 0.5. A linear initial velocity distribution between the boundary
points was used. The Thomas algorithm was employed to solve the resultant set
of simultaneous algebraic equations at each time step. The solution was
assumed to have reached a steady-state when the maximum change in u between
time steps was less than some specified value E.
RESULTS AND DISCUSSION
Nonconservative Form of Nonlinear Burgers' Equation
For v = 1124 (Rc = 2.4) and 
Cma-x 2Ax
At = 10, the solutions converged
using the Khosla-Rubin scheme in less than 50 steps to the steady-state
solution presented in Ref. 1 for e = 10 -5 . The solution is shown in
Figure 1. In the calculation, the nonlinear coefficient, a i , was lagged one
time-step and no iteration was performed at each time step. The solution was
a
5then continued further in an attempt to satisfy e = 10 -10 . After
approximately 30 more steps the solution began to diverge rapidly from the
previous result. (Converged results were obtained for e = 10 10 and large
values of C	 with the usual central difference form.) Similar behavior
max
was observed for Cmax = 0.9. For Cmax = 0.1, the solution converged in 741
steps for e = 10 10. In all cases the coefficient matrix is always diagonally
dominant. All test results given in the paper are summarized in Table I.
An effect of this instability in the solution can be seen in figure 2
where
^I (
ui+1	 "ui + 
ui-
l +
Di	 2Ax
is plotted for i - 26 (the midpoint) as a function of the number of time steps
for Cmax = 0.1, 0.9 and 10.0. This term, Di, is the quantity added to the
first-order (upwind) differencing of u  in the Khosla-Rubin modification and
appears on the right-hand side of (3), not as a coefficient of the unknowns.
As shown in figure 2, bi=26 remains small for Cm = 0.1;however, for
Cmax = 0.9 and 10.0, the term grows rapidly, and large errors appear in u.
Two types of iteration of the nonlinear term were also investigated for
the nonconservative form of Burgers' equation. The first of these was the
use of multiple iterations at each time step in the procedure previously
described. The case with v = 1/24 and C max = 0.9 was run with a fixed
number of iterations to update a  at each time step. However, the solutions
j	 again rapidly diverged with 2, 4, or 8 iterations per time step.
The second form of iteration used was Newton iteration (ref. 3). In this
technique
i
	 t
6
un+l,m+l _ un+l,m + du
	
(5)
where m is the iteration index a' each time step. Equation (5) is substi-
tuted into equation (2) to give tridiagonal equations of the form
Aidui-1 + B i Sui + Ci6ui
+1 = Di	 (6)
where, for ui 11/2
-V
Ai	 (AX)2
B = 1 
+ 1 (u'
n+1 - 
u
n+l,m )
 - 1 (un+19m - 1/2) + 2y
i 0t 2Ax i+l	 i-1	 bx i	 (AX) 2
Ci = 
-V 
2 + 1
— (ui+1 'm
 - 1/2)
(4x)
and for u  > 1/2,
Ai = 
-V 
2 - dx (ui
+1 'm
 - 1/2)
(11x)
B
i = 1 
+ 1 
(un+l,m - un+l,m ) + 1 (un+l,m - 112) + 2V
At 2Ax i+l	 i-1 	Dx i	 (Ax) 2
_ -V
C1 (^)2
I
In all cases,
ID = - 1 (un+l,m tin) + y (un+l,m - 2un+l,m + un+1,m)
	
i	 -At	 i	 i	 (AX) 2	 i+l	 i	 i-1
1 (un+l,m _ 1/2)(un+l,m - un+l'm)
20x i	 i+1	 i-1
It should be noted that tha? convective derivative in D i has been evaluated
using a central difference with values at the last iterate of the present time
step rather than with the usual Khosla-Rubin term. This modification was
found to be necessary to maintain the asymmetry of t2.e solution about the
midpoint.
Calculations were made in which the solution wa:i iterated at each time
step until the difference between iterations was less than 10
-10 . (The Newton
procedure could also be performed with no multiple iteration at each time
step, but such an approach was not used in the present investigation.) Thy
steady-state convergence criterion was again e = 10-10 . Converged solutions
were obtained for v = 1 /24 and Cmax - 0.9. However, the solution diverged
	
when Cm	was increased to 10.
Linear Burgers' Equation
To further demonstrate that the observed divergence is related to the
nonlinear term, tests were also made using a li:iear Burgers' equation
ut + aux = vuxx
	
(7)
A
where a is a constant. The bo^mdary conditions were again taken to be
t
!	 j
8
u(x,t) = 1.0 at x - -5
u(x,t) - 0 at x - +5
For this solution domain, the analytical solution is
eWX 
- 
e5w
u=
,-5w  - e5w
where w - a/v. For v - 1/24, equation (8) indicates that an extremely steep
gradient exists which could not be resolved with the coarse Ax = 0.2 grid;
therefore, v = 1 was used with values of a chosen such that R  > 2. Steady
state solutions are shown in figure 3 for two cases, R  - 2.4 and R  = 10.
Convergence was obtained in both cases for c v 10 -10 for values of C up
to 1000. The "wiggles" in the solution are not roundoff error but are the
exact algebraic solution of the difference equations.
Conservative Form of Nonlinear Burgers' Equation
Another way to treat the nonlinear term in equation (1) is to difference
the conservation form of Burgers' equation, i.e.,
ut +2 (u2 -u) x -vuxx 	(9)
The finite difference form of equation (9) is given by equation (2) where now
si 2 (ui+1 + ui_1 + 1) with Khosla-Rubin differencing.
Calculations were made in which a j was lagged one time step with no
multiple iteration performed at each time step. The steady-state convergence
10
criterion was again e = 10- . For v = 1/24 , converged solutions were
(8)
a
9obtained for all values of C
max 
used (0.9, 10, 100, and 1000). The solu-
tions are shown in figure 4. For v = 1/48 (Rc ,maX - 4.8), converged solutions
were obtained for C
max 
= 1000. When v was reduced to 1t96 (Rc max - 9.6),
solutions were obtained for Cmax = 0.9, but could not be obtained for
C___ = 10. The steady-state solutions for v - 1/48 and 1/96 are also shown
on figure 4. No other iteration schemes were tested with the Khosla-Rubin
method.
Cases were also run in which a central difference was used for the con-
vective term. (The procedure of ref. 1 was used with ai 
= 2 (ui+l + ui-1 + 1)
lagged one time step and without multiple iteration at each time step.) The
coefficient matrix is not diagonally dominant when R
	 > 2 unless
c ,max
Cmax < 1; however, converged solutions were obtained for v - 1/24 (Rc max 2.4)
and for v = 1/48 (Rc'max = 4.8) with values of Cmax up to 1000. Unlike the
Khosla-Rubin calculation for v - 1196, the solution converged for Cmax = 10,
although it diverged when C
max 
was 100.
Thus, for Burgers' equation with small values of v, the use of conserva-
tion form is appropriate. Since the solutions in such cases are steep waves,
this result is not unexpected.
CONCLUDING REMARKS
In previously reported implicit calculations of the nonlinear Burgers'
equation (in nonconservative form) using central finite differences for
spatial derivatives, solutions could not be obtained in many (but not all)
A	 cases when the cell Reynolds number was greater than 2. This failure was
attributed to the loss of diagonal dominance in the coefficient matrix which
occurs when Courant numbers greater than one are used for calculations in which
s
10
the maximum cell Reynolds number exceeds 2. The Khosla-Rubin modificatioa,
which gives an unconditionally diagonally dominant coefficient matrix, was
expected to give stable solutions for these cases. Numerical tests, however,
have shown that this method is unstable for cases in which the maximum cell
Reynolds number is greater than 2; i.e., for cases in which "wiggles" are
inherent in the solution to the difference equation, when the Courant number
is greater than 1. When the Courant number was less than 1, stable so--cations
could be obtained for some values of cell Reynolds number, but the maximum
allowable time step was found to be dependent upon tbi iteration technique
used. For the linear Purgers' equation, on the other t nd, no instability
was found even when "wiggles" were present. These rte^ultro suggest that the
failures may be the result of a nonlinear instability.
In the previous numerical tests with central differencing, nj cases were
found in which converged solutions could not be obtained when the maximum
cell Reynolds number was 2 or less. However, when the maximum cell Reynolds
number was greater than 2, solutions could not be obtained in some cases
even when the time step was small enough to give a diagonally dominant
coefficient matrix. Here again the feilure may have been the result of a
nonlinear instability.
This particular instability can be overcome for many cases by using the
conservative form of Burgers' equation for both the central difference and
Kaosla-Rubin metr-ds. The present study did not, however, attempt to
determine whether upper limits on the maximum cell Reynolds number and Courant
number exist for the conservative form. 	 S
i
	
it
t
2.1
RUTRENCES
1. Hirsh, Richard S., and Rudy, David H.: The Role of Diagonal Dominance and
Cell Reynolds Number in Implicit Difference Methods for Fluid Mechanics
Problems, Journal of Computational Physics, v. 16, no. 3. November 1974,
pp. 304-310.
2. Khosla, P. K.; and Rubin, S. 0.: A Dia,^Dnally Dominant Second-Order
Accurate Implicit Scheme. Computer and Fluids, v. 2, 1974,
pp. 207-209.
3. Keller, Herbert B.; and Cebeci, Tuncer; Accurate Numerical Methods for
Boundary-Layer Flows. II: Two-Dimensional Turbulent Flowe, AIP.A
Journal, v. 10, no. 9. September 1972, pp. 1193-1199.
•• m
•d
am
t,
a
rd
aM
$4
a
b
a
m
t+
rd
s
m
s.
d
am
1•i
a
•da
mN
10a
mk
Ida
$k4
d
am
$4
a
ba
m$4
^
am
k
a
ra
am
k
a
rd
am
k
a
rd
am
k
a
•d
a
m
s,
rd
vco
k
a
b
am
k
rd
am
Ir
a
rd
a
m
s^
d
am
$4
a
d
s
m
fr
s
a
a
>a >a a>>a >a a> a>>
a >
a >
a
4-1
>a >a >a >a >
a >'
a
°u
>
a >
0
a
-r-I
>a
°v
>
,~
°v°u v b ^ °u b b b °u °u u °u v
?S O O O a% H rn o^ m m o rn O O m O o O o O o 0
O H^ . r-+ o o • r+ o o H 0 0 0cpq
0r-I
O o 0 0 0 0 0 00 0 0 0
r-I 0 o
G7
0 1
W Ja
AW
7
G4
O
til
U]
H
N O
W +) •ri	 a^
w
u
w ^
s•,	 a
N .D
w N
b
4-i	 m	 O,^ O	 Iti •ri
O ^ O'
44 4>a	 01
N N N N N
	
N N N N C:	 :V N	 N OH
-7 -*	 .•? 7 _:r .7 --.t --.t --.r —t	 —t co %10 lD _:t co lZ %D r-I	 HN N N N N N N N N N N N .-I ON m N w* m m
O O O O O O O O O O O O O O O O O O O O
H U1	 r-I H ri ri ri H H H H	 r{1 1	 1 1 I 1 1 1 1 1 1	 1 1 1 I 1 I 1 I I	 1O O O O O O O O O O O O O O O O O 0 O O O
H ri	 H ri H ri H ri ri rI ri	 H ri ri H r-1 rI H H r-4	 ri
P4 04
P4 0
a a iti +31 CO N a a a aa a U) o a a >~
°a
v awN a z
.^ ao
r^-I •rOi r^ •ral r^-I •rai i r-i r^i
U
a
y Q! +)
OU ^ N
Nq a a 0U
IRWEDING PAGE BLANK— NOT
1.00
u
0.75
0, 50 0
1.2
	 1.60.4	 0.8
x
1
1
1.25 f-
Figure 1.- Steady-state solution for nonconservative form
of nonlinear Burgers' equation, v = 1/24, Rc,max = 2.4.
i
Figure 2.- Growth of D' term at midpoint of solution domain.
100	 200	 300	 400	 500	 600	 700
Time step
D26 10-?
10-9 0
10-5
10-6
10-8
u 
• 
1.8 
1.6 
1.4 
1.2 
a = 50, 
He = 10.0 
1.0r-------------------~~~~~~~~~-+~ 
0.8 
0.6 
0.4 
0.2 
a = 12, 
Re = 2.4 
0--------11~.0~------2t.~0------~3~.0~------4~.0~------~5.0 
x 
Figure 3.- Steady-state solution of linear Burgers' equations. 
1.30
1.10
U
0.90
0.70
0.50 0
0.4	 0.8
	 1.2
	 1.6
1
x
Figure 4.- Steady-state solution for conservative form of
nonlinear Burgers' equation.


Comments on the role of diagonal dominance in implicit difference methods

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server