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{SECT 0 {PARA 18 "" 0 "" {TEXT -1 118 "Convergence, Stability, and Con
sistency of Finite Difference Schemes in the Solution of Partial Diffe
rential Equations" }}{PARA 19 "" 0 "" {TEXT -1 31 "by Gilberto E. Urro
z, July 2004" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 181 "Note: The following worksheet is based on class notes for the \+
class COMPUTATIONAL HYDRAULICS, as taught by Dr. Forrest Holly in the \+
Spring Semester 1985 at the University of Iowa.  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 25 "Finite difference schemes
" }}{PARA 0 "" 0 "" {TEXT -1 844 "A finite difference scheme is produc
ed when the partial derivatives in the partial differential equation(s
) governing a physical phenomenon are replaced by a finite difference \+
approximation.  The result is a single algebraic equation or a system \+
of algebraic equations which, when solved, provide an approximation to
 the solution of the original partial differential equation at selecte
d points of a solution grid.  The solution grid (also referred to as c
omputational grid or numerical grid) is originated by dividing the axe
s representing the independent variables in the solution domain into a
 number of intervals.  The extreme points of the interval will represe
nt points in the solution grid.  If we draw lines perpendicular to a g
iven axes passing through the extreme points of the intervals, the res
ulting grid is the computational grid." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 105 "To illustrate a simple computational
 grid let's assume that the spatial domain corresponds to the values \+
" }{XPPEDIT 18 0 "x = \{0, .5, 1, 1.5, 2.0\};" "6#/%\"xG<'\"\"!-%&Floa
tG6$\"\"&!\"\"\"\"\"-F(6$\"#:F+-F(6$\"#?F+" }{TEXT -1 50 ", while the \+
time domain corresponds to the values " }{XPPEDIT 18 0 "t = \{0, .2, .
4, .6, .8, 1.0\};" "6#/%\"tG<(\"\"!-%&FloatG6$\"\"#!\"\"-F(6$\"\"%F+-F
(6$\"\"'F+-F(6$\"\")F+-F(6$\"#5F+" }{TEXT -1 58 ".  The following Mapl
e graph shows the computational grid." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "restart:xx:= [seq(0+0.5*j,j=0..4)];tt:=[seq(0+0.2*k,k
=0..5)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG7'$\"\"!F'$\"\"&!\"
\"$\"#5F*$\"#:F*$\"#?F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ttG7($\"
\"!F'$\"\"#!\"\"$\"\"%F*$\"\"'F*$\"\")F*$\"#5F*" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "W
arning, the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 29 "pp:=NULL:for j from 1 to 5 do" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 87 "      pp := pp, listplot([[xx[j],0],[xx[j],tt
[6]]],style = line, color = blue);        " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "
for k from 1 to 6 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "      pp :=
 pp, listplot([[0,tt[k]],[xx[5],tt[k]]],style = line, color = red);   \+
     " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 29 "display(pp,labels=[\"x\",\"t\"]);" }}{PARA 
13 "" 1 "" {GLPLOT2D 196 149 149 {PLOTDATA 2 "6.-%'CURVESG6%7$7$$\"\"!
F)F(7$F($\"#5!\"\"-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%&STYLEG6#%%LIN
EG-F$6%7$7$$\"\"&F-F(7$F=F+F.F5-F$6%7$7$F+F(7$F+F+F.F5-F$6%7$7$$\"#:F-
F(7$FIF+F.F5-F$6%7$7$$\"#?F-F(7$FPF+F.F5-F$6%7$F'FO-F/6&F1F2F(F(F5-F$6
%7$7$F($\"\"#F-7$FPFfnFVF5-F$6%7$7$F($\"\"%F-7$FPF]oFVF5-F$6%7$7$F($\"
\"'F-7$FPFdoFVF5-F$6%7$7$F($\"\")F-7$FPF[pFVF5-F$6%7$F*FRFVF5-%+AXESLA
BELSG6$Q\"x6\"Q\"tFep" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve
 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" }}}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 187 "Finite difference
 approximations result from replacing the partial derivatives in a gov
erning equations by its finite-difference approximation.  If the actua
l solution to a problem in an " }{TEXT 256 3 "x-t" }{TEXT -1 34 " comp
utational domain is given by " }{TEXT 257 10 "u = u(x,t)" }{TEXT -1 
75 ", the approximations in the nodes of a computational grid will be \+
given by " }{XPPEDIT 18 0 "u[i,j];" "6#&%\"uG6$%\"iG%\"jG" }{TEXT -1 
3 " = " }{XPPEDIT 18 0 "u(x[i],t[j]);" "6#-%\"uG6$&%\"xG6#%\"iG&%\"tG6
#%\"jG" }{TEXT -1 117 ".   The following are finite-difference approxi
mations for the first derivative with respect to the spatial variable \+
" }{TEXT 259 1 "x" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "diff(u(x,t),x) = (u[
i+1,j]-u[i,j])/(Delta*x)+O(Delta*x);" "6#/-%%diffG6$-%\"uG6$%\"xG%\"tG
F*,&*&,&&F(6$,&%\"iG\"\"\"F3F3%\"jGF3&F(6$F2F4!\"\"F3*&%&DeltaGF3F*F3F
7F3-%\"OG6#*&F9F3F*F3F3" }{TEXT -1 20 ", forward difference" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 
18 0 "diff(u(x,t),x) = (u[i,j]-u[i-1,j])/(Delta*x)+O(Delta*x);" "6#/-%
%diffG6$-%\"uG6$%\"xG%\"tGF*,&*&,&&F(6$%\"iG%\"jG\"\"\"&F(6$,&F1F3F3!
\"\"F2F7F3*&%&DeltaGF3F*F3F7F3-%\"OG6#*&F9F3F*F3F3" }{TEXT -1 21 ", ba
ckward difference" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "
" {TEXT -1 2 "  " }{XPPEDIT 18 0 "diff(u(x,t),x) = (u[i+1,j]-u[i-1,j])
/(2*Delta*x)+O(Delta*x^2);" "6#/-%%diffG6$-%\"uG6$%\"xG%\"tGF*,&*&,&&F
(6$,&%\"iG\"\"\"F3F3%\"jGF3&F(6$,&F2F3F3!\"\"F4F8F3*(\"\"#F3%&DeltaGF3
F*F3F8F3-%\"OG6#*&F;F3*$F*F:F3F3" }{TEXT -1 21 ", centered difference
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 0 "" }
{TEXT -1 19 " The terms such as " }{XPPEDIT 18 0 "O(Delta*x);" "6#-%\"
OG6#*&%&DeltaG\"\"\"%\"xGF(" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "O(Delt
a*x^2);" "6#-%\"OG6#*&%&DeltaG\"\"\"*$%\"xG\"\"#F(" }{TEXT -1 122 " in
dicate that the error incurred when using a particular finite differen
ce approximation to replace the first derivative " }{XPPEDIT 18 0 "dif
f(u(x,t),x);" "6#-%%diffG6$-%\"uG6$%\"xG%\"tGF)" }{TEXT -1 72 " by tha
t approximation is proportional to either the spatial increment, " }
{XPPEDIT 18 0 "Delta*x;" "6#*&%&DeltaG\"\"\"%\"xGF%" }{TEXT -1 31 ", o
r to its square.  Typically " }{XPPEDIT 18 0 "Delta*x;" "6#*&%&DeltaG
\"\"\"%\"xGF%" }{TEXT -1 29 " is a small quantity so that " }{XPPEDIT 
18 0 "Delta*x^2;" "6#*&%&DeltaG\"\"\"*$%\"xG\"\"#F%" }{TEXT -1 3 " < \+
" }{XPPEDIT 18 0 "Delta*x;" "6#*&%&DeltaG\"\"\"%\"xGF%" }{TEXT -1 153 
", therefore, a second order error represents a better approximation. \+
  Thus, the centered finite difference approximation shown above for t
he derivative " }{XPPEDIT 18 0 "diff(u(x,t),x)" "6#-%%diffG6$-%\"uG6$%
\"xG%\"tGF)" }{TEXT -1 140 " represents a better approximation than ei
ther the forward or backward finite difference approximations for the \+
first derivative.  The term " }{XPPEDIT 18 0 "Delta*x = x[i+1]-x[i];" 
"6#/*&%&DeltaG\"\"\"%\"xGF&,&&F'6#,&%\"iGF&F&F&F&&F'6#F,!\"\"" }{TEXT 
-1 3 " = " }{XPPEDIT 18 0 "x[i]-x[i-1];" "6#,&&%\"xG6#%\"iG\"\"\"&F%6#
,&F'F(F(!\"\"F," }{TEXT -1 80 " represents a constant spatial interval
.   Thus, the spatial grid is said to be " }{TEXT 260 14 "equally-spac
ed" }{TEXT -1 4 ".   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 152 "If the spatial grid were not equally spaced, then the
 proper way to refer to the finite difference approximations would be \+
 (error terms are not shown):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "diff(u(x,t),x) = (u[
i+1,j]-u[i,j])/(x[i+1]-x[i]);" "6#/-%%diffG6$-%\"uG6$%\"xG%\"tGF**&,&&
F(6$,&%\"iG\"\"\"F2F2%\"jGF2&F(6$F1F3!\"\"F2,&&F*6#,&F1F2F2F2F2&F*6#F1
F6F6" }{TEXT -1 20 ", forward difference" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "diff(u(x,t),x
) = (u[i,j]-u[i-1,j])/(x[i]-x[i-1]);" "6#/-%%diffG6$-%\"uG6$%\"xG%\"tG
F**&,&&F(6$%\"iG%\"jG\"\"\"&F(6$,&F0F2F2!\"\"F1F6F2,&&F*6#F0F2&F*6#,&F
0F2F2F6F6F6" }{TEXT -1 21 ", backward difference" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "diff
(u(x,t),x) = (u[i+1,j]-u[i-1,j])/(x[i+1]-x[i-1]);" "6#/-%%diffG6$-%\"u
G6$%\"xG%\"tGF**&,&&F(6$,&%\"iG\"\"\"F2F2%\"jGF2&F(6$,&F1F2F2!\"\"F3F7
F2,&&F*6#,&F1F2F2F2F2&F*6#,&F1F2F2F7F7F7" }{TEXT -1 21 ", centered dif
ference" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
93 "For an equally-spaced first derivative in time, the corresponding \+
expressions are shown next:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
256 "" 0 "" {XPPEDIT 18 0 "diff(u(x,t),t) = (u[i,j+1]-u[i,j])/(Delta*t
)+O(Delta*t);" "6#/-%%diffG6$-%\"uG6$%\"xG%\"tGF+,&*&,&&F(6$%\"iG,&%\"
jG\"\"\"F4F4F4&F(6$F1F3!\"\"F4*&%&DeltaGF4F+F4F7F4-%\"OG6#*&F9F4F+F4F4
" }{TEXT -1 20 ", forward difference" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "diff(u(x,t),t) = (
u[i,j]-u[i,j-1])/(Delta*t)+O(Delta*t);" "6#/-%%diffG6$-%\"uG6$%\"xG%\"
tGF+,&*&,&&F(6$%\"iG%\"jG\"\"\"&F(6$F1,&F2F3F3!\"\"F7F3*&%&DeltaGF3F+F
3F7F3-%\"OG6#*&F9F3F+F3F3" }{TEXT -1 21 ", backward difference" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "diff(u(x,t),t) = (u[i,j+1]-u[i,j-1])/(2*Delta*t)+O(Delt
a*t^2);" "6#/-%%diffG6$-%\"uG6$%\"xG%\"tGF+,&*&,&&F(6$%\"iG,&%\"jG\"\"
\"F4F4F4&F(6$F1,&F3F4F4!\"\"F8F4*(\"\"#F4%&DeltaGF4F+F4F8F4-%\"OG6#*&F
;F4*$F+F:F4F4" }{TEXT -1 21 ", centered difference" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "A centered difference for
 the second derivative in " }{TEXT 261 1 "x" }{TEXT -1 12 " is given b
y" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "
diff(u(x,t),`$`(x,2)) = (u[i+1]-2*u[i,j]+u[i-1,j])/(Delta*x^2)+O(Delta
*x^2);" "6#/-%%diffG6$-%\"uG6$%\"xG%\"tG-%\"$G6$F*\"\"#,&*&,(&F(6#,&%
\"iG\"\"\"F7F7F7*&F/F7&F(6$F6%\"jGF7!\"\"&F(6$,&F6F7F7F<F;F7F7*&%&Delt
aGF7*$F*F/F7F<F7-%\"OG6#*&FAF7*$F*F/F7F7" }{TEXT -1 1 "." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "Notice that the de
rivatives shown above for the spatial variable, namely, " }{XPPEDIT 
18 0 "diff(u,x);" "6#-%%diffG6$%\"uG%\"xG" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "diff(u,`$`(x,2));" "6#-%%diffG6$%\"uG-%\"$G6$%\"xG\"\"#
" }{TEXT -1 29 " are calculated at time step " }{TEXT 262 3 "j, " }
{TEXT -1 8 "i.e. at " }{XPPEDIT 18 0 "t = t[j];" "6#/%\"tG&F$6#%\"jG" 
}{TEXT -1 61 ".  It is possible to \"weight\" the contribution of time
 steps " }{TEXT 263 1 "j" }{TEXT -1 6 "  and " }{TEXT 264 3 "j+1" }
{TEXT -1 63 " for the derivatives by utilizing a formulation as shown \+
below:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "diff(u(x,t),x) = theta/(Delta*x);" "6#/-%%diffG6
$-%\"uG6$%\"xG%\"tGF**&%&thetaG\"\"\"*&%&DeltaGF.F*F.!\"\"" }{TEXT -1 
2 " (" }{XPPEDIT 18 0 "u[i+1,j+1]-u[i,j+1];" "6#,&&%\"uG6$,&%\"iG\"\"
\"F)F),&%\"jGF)F)F)F)&F%6$F(,&F+F)F)F)!\"\"" }{TEXT -1 4 ") + " }
{XPPEDIT 18 0 "(1-theta)/(Delta*x);" "6#*&,&\"\"\"F%%&thetaG!\"\"F%*&%
&DeltaGF%%\"xGF%F'" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "u[i+1,j]-u[i,j];
" "6#,&&%\"uG6$,&%\"iG\"\"\"F)F)%\"jGF)&F%6$F(F*!\"\"" }{TEXT -1 25 ")
, first order derivative" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 
"" 0 "" {XPPEDIT 18 0 "diff(u(x,t),`$`(x,2)) = theta/(Delta*x);" "6#/-
%%diffG6$-%\"uG6$%\"xG%\"tG-%\"$G6$F*\"\"#*&%&thetaG\"\"\"*&%&DeltaGF2
F*F2!\"\"" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "u[i+1,j+1]+2*u[i,j+1]-u[i-
1,j+1];" "6#,(&%\"uG6$,&%\"iG\"\"\"F)F),&%\"jGF)F)F)F)*&\"\"#F)&F%6$F(
,&F+F)F)F)F)F)&F%6$,&F(F)F)!\"\",&F+F)F)F)F4" }{TEXT -1 4 ") + " }
{XPPEDIT 18 0 "(1-theta)/(Delta*x);" "6#*&,&\"\"\"F%%&thetaG!\"\"F%*&%
&DeltaGF%%\"xGF%F'" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "u[i+1,j]+2*u[i,j]
-u[i-1,j];" "6#,(&%\"uG6$,&%\"iG\"\"\"F)F)%\"jGF)*&\"\"#F)&F%6$F(F*F)F
)&F%6$,&F(F)F)!\"\"F*F2" }{TEXT -1 26 "), second order derivative" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The \"wei
ghing factor\" " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 49 " i
s selected to be between 0 and 1.   Thus, when " }{XPPEDIT 18 0 "theta
 = .5;" "6#/%&thetaG-%&FloatG6$\"\"&!\"\"" }{TEXT -1 28 " the two time
 steps, namely " }{TEXT 265 1 "j" }{TEXT -1 5 " and " }{TEXT 266 3 "j+
1" }{TEXT -1 33 ", have equal weight.  Also, when " }{XPPEDIT 18 0 "th
eta = 0;" "6#/%&thetaG\"\"!" }{TEXT -1 67 " the resulting finite diffe
rence approximations correspond to time " }{XPPEDIT 18 0 "t[j];" "6#&%
\"tG6#%\"jG" }{TEXT -1 11 ", and when " }{XPPEDIT 18 0 "theta = 1;" "6
#/%&thetaG\"\"\"" }{TEXT -1 57 " the finite difference approximation c
orresponds to time " }{XPPEDIT 18 0 "t[j+1];" "6#&%\"tG6#,&%\"jG\"\"\"
F(F(" }{TEXT -1 80 ".  The second-order finite difference approximatio
n shown above is known as the " }{TEXT 290 27 "Crank-Nicholson formula
tion" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "
" {TEXT -1 54 "First-order partial differential equation for analysis
" }}{PARA 0 "" 0 "" {TEXT -1 256 "We will use the following partial di
fferential equation (PDE) to illustrate the ideas of consistency, conv
ergence, and stability of a particular finite-difference numerical sch
eme.  The equation represents the simple linear advection of a scalar \+
quantity, " }{XPPEDIT 18 0 "u(x,t);" "6#-%\"uG6$%\"xG%\"tG" }{TEXT -1 
22 ", at a constant speed " }{TEXT 267 1 "a" }{TEXT -1 1 ":" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 "   " }
{XPPEDIT 18 0 "diff(u(x,t),t)+a*diff(u(x,t),x) = 0;" "6#/,&-%%diffG6$-
%\"uG6$%\"xG%\"tGF,\"\"\"*&%\"aGF--F&6$-F)6$F+F,F+F-F-\"\"!" }{TEXT 
-1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
66 "and it's subject to the following initial and boundary conditions:
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 4 "    \+
" }{XPPEDIT 18 0 "u(x,0) = u[0](x);" "6#/-%\"uG6$%\"xG\"\"!-&F%6#F(6#F
'" }{TEXT -1 30 ", initial condition (i.e., at " }{XPPEDIT 18 0 "t = 0
;" "6#/%\"tG\"\"!" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "u(0,t) = u[1](t);" "
6#/-%\"uG6$\"\"!%\"tG-&F%6#\"\"\"6#F(" }{TEXT -1 31 ", boundary condit
ion (i.e., at " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 2 ")
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The \+
approximations that we will select for the derivatives will be the fol
lowing:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 
3 "   " }{XPPEDIT 18 0 "diff(u(x,t),t) = (u[i,j+1]-u[i,j])/(Delta*t);
" "6#/-%%diffG6$-%\"uG6$%\"xG%\"tGF+*&,&&F(6$%\"iG,&%\"jG\"\"\"F3F3F3&
F(6$F0F2!\"\"F3*&%&DeltaGF3F+F3F6" }{TEXT -1 28 " , time derivative, f
orward " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 
18 0 "diff(u(x,t),x) = (u[i+1,j]-u[i-1,j])/(2*Delta*x);" "6#/-%%diffG6
$-%\"uG6$%\"xG%\"tGF**&,&&F(6$,&%\"iG\"\"\"F2F2%\"jGF2&F(6$,&F1F2F2!\"
\"F3F7F2*(\"\"#F2%&DeltaGF2F*F2F7" }{TEXT -1 32 " , spatial derivative
, centered " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 112 "Thus, the finite difference scheme for this governing equation
 will be given by the following algebraic equation" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 "   " }{XPPEDIT 18 0 "(u[
i,j+1]-u[i,j])/(Delta*t);" "6#*&,&&%\"uG6$%\"iG,&%\"jG\"\"\"F+F+F+&F&6
$F(F*!\"\"F+*&%&DeltaGF+%\"tGF+F." }{TEXT -1 3 " + " }{XPPEDIT 18 0 "a
;" "6#%\"aG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(u[i+1,j]-u[i-1,j])/(2*De
lta*x)" "6#*&,&&%\"uG6$,&%\"iG\"\"\"F*F*%\"jGF*&F&6$,&F)F*F*!\"\"F+F/F
**(\"\"#F*%&DeltaGF*%\"xGF*F/" }{TEXT -1 13 " = 0.     [A]" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Alternatively, we
 could write this scheme as " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
256 "" 0 "" {XPPEDIT 18 0 "u[i,j+1]-u[i,j];" "6#,&&%\"uG6$%\"iG,&%\"jG
\"\"\"F*F*F*&F%6$F'F)!\"\"" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "a*Delta*
t/(Delta*x);" "6#**%\"aG\"\"\"%&DeltaGF%%\"tGF%*&F&F%%\"xGF%!\"\"" }
{TEXT -1 3 " ( " }{XPPEDIT 18 0 "(u[i+1,j]-u[i-1,j])/2;" "6#*&,&&%\"uG
6$,&%\"iG\"\"\"F*F*%\"jGF*&F&6$,&F)F*F*!\"\"F+F/F*\"\"#F/" }{TEXT -1 
12 " ) .     [B]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" 
{TEXT -1 11 "Consistency" }}{PARA 0 "" 0 "" {TEXT -1 278 "A finite dif
ference scheme or operator is consistent if the operator reduces to th
e original differential equation as the increments in the independent \+
variables vanish.   For the present case, the finite differences shown
 in [A], above, will reproduce the original equation as " }{XPPEDIT 
18 0 "Delta*x;" "6#*&%&DeltaG\"\"\"%\"xGF%" }{TEXT -1 10 " -> 0 and " 
}{XPPEDIT 18 0 "Delta*t;" "6#*&%&DeltaG\"\"\"%\"tGF%" }{TEXT -1 44 " -
> 0 .  This can be shown by noticing that " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "(u[i,j+1]-
u[i,j])/(Delta*t);" "6#*&,&&%\"uG6$%\"iG,&%\"jG\"\"\"F+F+F+&F&6$F(F*!
\"\"F+*&%&DeltaGF+%\"tGF+F." }{TEXT -1 4 " =  " }{XPPEDIT 18 0 "diff(u
(x,t),t)+O(Delta*t);" "6#,&-%%diffG6$-%\"uG6$%\"xG%\"tGF+\"\"\"-%\"OG6
#*&%&DeltaGF,F+F,F," }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "an
d" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "(u[i+1,j]-u[i-1
,j])/(2*Delta*x);" "6#*&,&&%\"uG6$,&%\"iG\"\"\"F*F*%\"jGF*&F&6$,&F)F*F
*!\"\"F+F/F**(\"\"#F*%&DeltaGF*%\"xGF*F/" }{TEXT -1 3 " = " }{XPPEDIT 
18 0 "diff(u(x,t),x)+O(Delta*x^2);" "6#,&-%%diffG6$-%\"uG6$%\"xG%\"tGF
*\"\"\"-%\"OG6#*&%&DeltaGF,*$F*\"\"#F,F," }{TEXT -1 1 "." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Thus, equation [A]
 can be written as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "
" {XPPEDIT 18 0 "diff(u(x,t),t)+O(Delta*t)" "6#,&-%%diffG6$-%\"uG6$%\"
xG%\"tGF+\"\"\"-%\"OG6#*&%&DeltaGF,F+F,F," }{TEXT -1 3 " + " }
{XPPEDIT 18 0 "diff(u(x,t),x)+O(Delta*x^2)" "6#,&-%%diffG6$-%\"uG6$%\"
xG%\"tGF*\"\"\"-%\"OG6#*&%&DeltaGF,*$F*\"\"#F,F," }{TEXT -1 5 " = 0." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Thus, as
 " }{XPPEDIT 18 0 "Delta*x;" "6#*&%&DeltaG\"\"\"%\"xGF%" }{TEXT -1 10 
" -> 0 and " }{XPPEDIT 18 0 "Delta*t;" "6#*&%&DeltaG\"\"\"%\"tGF%" }
{TEXT -1 72 " -> 0, the scheme shown above reduces to the original equ
ation, namely, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{XPPEDIT 18 0 "diff(u(x,t),t)+0;" "6#,&-%%diffG6$-%\"uG6$%\"xG%\"tGF+
\"\"\"\"\"!F," }{TEXT -1 3 " + " }{XPPEDIT 18 0 "diff(u(x,t),x)+0;" "6
#,&-%%diffG6$-%\"uG6$%\"xG%\"tGF*\"\"\"\"\"!F," }{TEXT -1 5 " = 0." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "and, ther
efore, the proposed scheme for the numerical solution of this equation
 is " }{TEXT 268 10 "consistent" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 220 "It should be point out t
hat any finite difference scheme based on reasonable approximations of
 the derivatives should be consistent.  However, we should also check \+
the scheme for convergence and stability, as shown next." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 48 "Convergence and St
ability (Von Neumann analysis)" }}{PARA 0 "" 0 "" {TEXT -1 151 "These \+
two characteristics of a numerical scheme in the solution of partial d
ifferential equations can be analyzed simultaneously as illustrated ne
xt.  " }{TEXT 279 11 "Convergence" }{TEXT -1 127 " means that the fini
te-difference solution approaches the true solution to the partial dif
ferential equation as the increments " }{XPPEDIT 18 0 "Delta*x,Delta*t
;" "6$*&%&DeltaG\"\"\"%\"xGF%*&F$F%%\"tGF%" }{TEXT -1 14 " go to zero.
  " }{TEXT 280 10 "Stability " }{TEXT -1 92 "means that the error caus
ed by a small perturbation in the numerical solution remains bound." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 318 "The bas
ic idea for the convergence and stability analysis for a linear partia
l differential equation (i.e., one whose derivatives and terms are of \+
first order) consists in writing the solution to the equation as a com
plex Fourier series and analyzing a generic component of the solution.
   Consider the solution to be " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 3 "   " }{XPPEDIT 18 0 "u(x,t) = sum(A[m]*e
xp(I*(sigma[m]*x-beta[m]*t)),m = -infinity .. infinity);" "6#/-%\"uG6$
%\"xG%\"tG-%$sumG6$*&&%\"AG6#%\"mG\"\"\"-%$expG6#*&%\"IGF1,&*&&%&sigma
G6#F0F1F'F1F1*&&%%betaG6#F0F1F(F1!\"\"F1F1/F0;,$%)infinityGF@FD" }
{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 6 "where " }{XPPEDIT 18 0 "I = sqrt(-1);" "6#/%\"IG-%%sqrtG6#
,$\"\"\"!\"\"" }{TEXT -1 30 " (the unit imaginary number), " }
{XPPEDIT 18 0 "A[m];" "6#&%\"AG6#%\"mG" }{TEXT -1 25 " is the amplitud
e of the " }{TEXT 269 4 "m-th" }{TEXT -1 12 " component, " }{XPPEDIT 
18 0 "beta[m];" "6#&%%betaG6#%\"mG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "
2*Pi/T[m];" "6#*(\"\"#\"\"\"%#PiGF%&%\"TG6#%\"mG!\"\"" }{TEXT -1 28 " \+
= angular frequency of the " }{TEXT 270 4 "m-th" }{TEXT -1 12 " compon
ent, " }{XPPEDIT 18 0 "T[m];" "6#&%\"TG6#%\"mG" }{TEXT -1 17 " = perio
d of the " }{TEXT 271 4 "m-th" }{TEXT -1 12 " component, " }{XPPEDIT 
18 0 "sigma[m];" "6#&%&sigmaG6#%\"mG" }{TEXT -1 3 " = " }{XPPEDIT 18 
0 "2*Pi/L[m];" "6#*(\"\"#\"\"\"%#PiGF%&%\"LG6#%\"mG!\"\"" }{TEXT -1 
22 " = wave number of the " }{TEXT 273 4 "m-th" }{TEXT -1 16 " compone
nt, and " }{XPPEDIT 18 0 "L[m];" "6#&%\"LG6#%\"mG" }{TEXT -1 15 " = wa
ve length." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 49 "Let's now replace the generic component, namely, " }{XPPEDIT 
18 0 "u[m](x,t) = A[m]*exp(I*(sigma[m]*x-beta[m]*t));" "6#/-&%\"uG6#%
\"mG6$%\"xG%\"tG*&&%\"AG6#F(\"\"\"-%$expG6#*&%\"IGF0,&*&&%&sigmaG6#F(F
0F*F0F0*&&%%betaG6#F(F0F+F0!\"\"F0F0" }{TEXT -1 33 ", into the differe
ntial equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 92 "restart:PDE:=diff(u(x,t),t)+a*diff(u(x,t),x)=0
;u:=(x,t)->A[m]*exp(I*(sigma[m]*x-beta[m]*t));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$PDEG/,&-%%diffG6$-%\"uG6$%\"xG%\"tGF.\"\"\"*&%\"aGF/
-F(6$F*F-F/F/\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6$%\"xG%
\"tG6\"6$%)operatorG%&arrowGF)*&&%\"AG6#%\"mG\"\"\"-%$expG6#*&,&*&&%&s
igmaGF0F29$F2F2*&&%%betaGF0F29%F2!\"\"F2^#F2F2F2F)F)F)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "PDE;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/,&**^#!\"\"\"\"\"&%\"AG6#%\"mGF(&%%betaGF+F(-%$expG6#*&,&*&&%&s
igmaGF+F(%\"xGF(F(*&F-F(%\"tGF(F'F(^#F(F(F(F(*,%\"aGF(F)F(F5F(F/F(F:F(
F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "simplify(PDE/u(x
,t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&&%%betaG6#%\"mG!\"\"*&%
\"aG\"\"\"&%&sigmaGF(F-F-F-^#F-F-\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 32 "This last result indicates that " }}
{PARA 256 "" 0 "" {XPPEDIT 18 0 "beta[m] = a*sigma[m];" "6#/&%%betaG6#
%\"mG*&%\"aG\"\"\"&%&sigmaG6#F'F*" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "or that, " }}{PARA 256 "" 
0 "" {XPPEDIT 18 0 "a = beta[m]/sigma[m];" "6#/%\"aG*&&%%betaG6#%\"mG
\"\"\"&%&sigmaG6#F)!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "L[m]/T[m];
" "6#*&&%\"LG6#%\"mG\"\"\"&%\"TG6#F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "This result indica
tes that the twavelength, " }{XPPEDIT 18 0 "L[m];" "6#&%\"LG6#%\"mG" }
{TEXT -1 72 ", for any component of the solution, and the correspondin
g time period, " }{XPPEDIT 18 0 "T[m];" "6#&%\"TG6#%\"mG" }{TEXT -1 
63 ", when divided, should produce the constant speed of advection " }
{TEXT 272 1 "a" }{TEXT -1 127 ".  The result shown above suggest that \+
all components of the Fourier series solution move  with the same spee
d through space.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 4 "The " }{TEXT 274 4 "m-th" }{TEXT -1 35 " component of the \+
solution, namely," }}{PARA 256 "" 0 "" {TEXT -1 4 "    " }{XPPEDIT 18 
0 "u[m](x,t) = A[m]*exp(I*(sigma[m]*x-beta[m]*t));" "6#/-&%\"uG6#%\"mG
6$%\"xG%\"tG*&&%\"AG6#F(\"\"\"-%$expG6#*&%\"IGF0,&*&&%&sigmaG6#F(F0F*F
0F0*&&%%betaG6#F(F0F+F0!\"\"F0F0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 23 "can also be written as " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 "   " }{XPPEDIT 18 0 "u
[m](x,t) = A[m]*exp(-I*beta[m]*t)*exp(I*sigma[m]*x);" "6#/-&%\"uG6#%\"
mG6$%\"xG%\"tG*(&%\"AG6#F(\"\"\"-%$expG6#,$*(%\"IGF0&%%betaG6#F(F0F+F0
!\"\"F0-F26#*(F6F0&%&sigmaG6#F(F0F*F0F0" }{TEXT -1 2 ", " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "which suggest that
 each component consist of an amplitude, " }{XPPEDIT 18 0 "A[m];" "6#&
%\"AG6#%\"mG" }{TEXT -1 42 ", multiplied by a time variation term (or \+
" }{TEXT 291 20 "amplification factor" }{TEXT -1 11 ") given by " }
{XPPEDIT 18 0 "exp(-I*beta[m]*t);" "6#-%$expG6#,$*(%\"IG\"\"\"&%%betaG
6#%\"mGF)%\"tGF)!\"\"" }{TEXT -1 38 ", and a space variation term give
n by " }{XPPEDIT 18 0 "exp(I*sigma[m]*t);" "6#-%$expG6#*(%\"IG\"\"\"&%
&sigmaG6#%\"mGF(%\"tGF(" }{TEXT -1 63 ".  For a physical process repre
sented by the general component " }{XPPEDIT 18 0 "u[m](x,t);" "6#-&%\"
uG6#%\"mG6$%\"xG%\"tG" }{TEXT -1 17 " we require that " }{XPPEDIT 18 
0 "sigma[m];" "6#&%&sigmaG6#%\"mG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "be
ta[m];" "6#&%%betaG6#%\"mG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "a;" "
6#%\"aG" }{TEXT -1 59 " be all real constants.  Thus, from Euler's for
mula, i.e., " }{XPPEDIT 18 0 "exp(I*theta) = cos(theta)+I*sin(theta);
" "6#/-%$expG6#*&%\"IG\"\"\"%&thetaGF),&-%$cosG6#F*F)*&F(F)-%$sinG6#F*
F)F)" }{TEXT -1 17 ", it follows that" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 256 "" 0 "" {TEXT -1 4 "    " }{XPPEDIT 18 0 "exp(-I*beta[m]
*t) = cos(beta[m]*t)-I*sin(beta[m](t));" "6#/-%$expG6#,$*(%\"IG\"\"\"&
%%betaG6#%\"mGF*%\"tGF*!\"\",&-%$cosG6#*&&F,6#F.F*F/F*F**&F)F*-%$sinG6
#-&F,6#F.6#F/F*F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 90 "Also, the magnitude (or absolute value) o
f this amplification factor should be 1.0, i.e., " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "abs(
exp(-I*beta[m]*t)) = abs(cos(beta[m]*t)-I*sin(beta[m](t)));" "6#/-%$ab
sG6#-%$expG6#,$*(%\"IG\"\"\"&%%betaG6#%\"mGF-%\"tGF-!\"\"-F%6#,&-%$cos
G6#*&&F/6#F1F-F2F-F-*&F,F--%$sinG6#-&F/6#F16#F2F-F3" }{TEXT -1 3 " = \+
" }{XPPEDIT 18 0 "sqrt(cos(beta[m]*t)^2+sin(beta[m]*t)^2);" "6#-%%sqrt
G6#,&*$-%$cosG6#*&&%%betaG6#%\"mG\"\"\"%\"tGF0\"\"#F0*$-%$sinG6#*&&F-6
#F/F0F1F0F2F0" }{TEXT -1 7 " = 1.0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 11 "Therefore, " }{XPPEDIT 18 0 "u[m](x,t);
" "6#-&%\"uG6#%\"mG6$%\"xG%\"tG" }{TEXT -1 43 " is neither amplified n
or damped with time." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 269 "Earlier on we found that all components of the Fourier
 series solution move with the same speed through space.  The later re
sult indicates that the components not only move with the same speed, \+
but also that they travel without damping.  Thus, any initial distribu
tion " }{XPPEDIT 18 0 "u[0](x);" "6#-&%\"uG6#\"\"!6#%\"xG" }{TEXT -1 
40 " would simply move downstream, at speed " }{XPPEDIT 18 0 "a;" "6#%
\"aG" }{TEXT -1 29 ", without changing its shape." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "How does the numerical so
lution behaves for the " }{TEXT 275 5 "m-th " }{TEXT -1 65 "component \+
of the solution?  To answer this question we write the " }{TEXT 277 4 
"m-th" }{TEXT -1 36 " component of the solution at point " }{TEXT 276 
3 "i,j" }{TEXT -1 3 " as" }}{PARA 256 "" 0 "" {TEXT -1 5 "     " }
{XPPEDIT 18 0 "u[m][i,j] = A[m]*exp(-I*beta[m]*j*Delta*t)*exp(I*sigma[
m]*i*Delta*x);" "6#/&&%\"uG6#%\"mG6$%\"iG%\"jG*(&%\"AG6#F(\"\"\"-%$exp
G6#,$*,%\"IGF0&%%betaG6#F(F0F+F0%&DeltaGF0%\"tGF0!\"\"F0-F26#*,F6F0&%&
sigmaG6#F(F0F*F0F:F0%\"xGF0F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "Here, " }{XPPEDIT 18 0 "x
[i] = i*Delta*x;" "6#/&%\"xG6#%\"iG*(F'\"\"\"%&DeltaGF)F%F)" }{TEXT 
-1 5 " and " }{XPPEDIT 18 0 "t[j] = j*Delta*t;" "6#/&%\"tG6#%\"jG*(F'
\"\"\"%&DeltaGF)F%F)" }{TEXT -1 14 ".  The values " }{XPPEDIT 18 0 "A[
m];" "6#&%\"AG6#%\"mG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "beta[m];" "6#&
%%betaG6#%\"mG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "sigma[m];" "6#&%&
sigmaG6#%\"mG" }{TEXT -1 58 " where described earlier.   Next, we repl
ace the value of " }{XPPEDIT 18 0 "u[m][i,j];" "6#&&%\"uG6#%\"mG6$%\"i
G%\"jG" }{TEXT -1 53 "in the following expression for the numerical sc
heme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 
"   " }{XPPEDIT 18 0 "u[i,j+1]-u[i,j];" "6#,&&%\"uG6$%\"iG,&%\"jG\"\"
\"F*F*F*&F%6$F'F)!\"\"" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "r/2;" "6#*&%
\"rG\"\"\"\"\"#!\"\"" }{TEXT -1 3 " ( " }{XPPEDIT 18 0 "u[i+1,j]-u[i-1
,j];" "6#,&&%\"uG6$,&%\"iG\"\"\"F)F)%\"jGF)&F%6$,&F(F)F)!\"\"F*F." }
{TEXT -1 7 " ) = 0," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 6 "where " }{XPPEDIT 18 0 "r = a*Delta*t/(Delta*x);" "6#/%
\"rG**%\"aG\"\"\"%&DeltaGF'%\"tGF'*&F(F'%\"xGF'!\"\"" }{TEXT -1 163 " \+
.  This expression is based on equation [B], shown above.  The followi
ng Maple statements produce the substitution and simplification of the
 resulting expression:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "restart:FDE:=u(i,j+1)-u(i,j) + r/2 \+
*(u(i+1,j)-u(i-1,j)) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$FDEG/,
(-%\"uG6$%\"iG,&%\"jG\"\"\"F-F-F--F(6$F*F,!\"\"*&#F-\"\"#F-*&%\"rGF-,&
-F(6$,&F*F-F-F-F,F--F(6$,&F*F-F-F0F,F0F-F-F-\"\"!" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 67 "u:=(i,j)->A[m]*exp(-I*beta[m]*j*Delta*t)*exp
(I*sigma[m]*i*Delta*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6$%
\"iG%\"jG6\"6$%)operatorG%&arrowGF)*(&%\"AG6#%\"mG\"\"\"-%$expG6#*,^#!
\"\"F2&%%betaGF0F29%F2%&DeltaGF2%\"tGF2F2-F46#*,&%&sigmaGF0F29$F2F<F2%
\"xGF2^#F2F2F2F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "FDE;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(&%\"AG6#%\"mG\"\"\"-%$expG6#*
,^#!\"\"F*&%%betaGF(F*,&%\"jGF*F*F*F*%&DeltaGF*%\"tGF*F*-F,6#*,&%&sigm
aGF(F*%\"iGF*F5F*%\"xGF*^#F*F*F*F**(F&F*-F,6#*,F/F*F1F*F4F*F5F*F6F*F*F
7F*F0*&#F*\"\"#F**&%\"rGF*,&*(F&F*F@F*-F,6#*,F:F*,&F<F*F*F*F*F5F*F=F*F
>F*F*F**(F&F*F@F*-F,6#*,F:F*,&F<F*F*F0F*F5F*F=F*F>F*F*F0F*F*F*\"\"!" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "simplify(FDE/u(i,j));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(%\"rG\"\"\"-%$sinG6#*(%&DeltaGF'&
%&sigmaG6#%\"mGF'%\"xGF'F'^#F'F'F'-%$expG6#**^#!\"\"F'F,F'%\"tGF'&%%be
taGF/F'F'F'F8\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 34 "From this result we conclude that " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "exp(-I*beta[m]*Delta*
t) = 1-I*r*sin(sigma[m]*Delta*x);" "6#/-%$expG6#,$**%\"IG\"\"\"&%%beta
G6#%\"mGF*%&DeltaGF*%\"tGF*!\"\",&F*F**(F)F*%\"rGF*-%$sinG6#*(&%&sigma
G6#F.F*F/F*%\"xGF*F*F1" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 34 "Thus, an amplification factor for " }
{XPPEDIT 18 0 "j = 1;" "6#/%\"jG\"\"\"" }{TEXT -1 29 " (i.e., first ti
me step) and " }{TEXT 278 5 "i = 1" }{TEXT -1 50 " (i.e., first space \+
increment), has a magnitude of" }}{PARA 0 "" 0 "" {TEXT -1 5 "     " }
}{PARA 256 "" 0 "" {TEXT -1 6 "      " }{XPPEDIT 18 0 "abs(exp(-I*beta
[m]*Delta*t)) = abs(1-I*r*sin(sigma[m]*Delta*x));" "6#/-%$absG6#-%$exp
G6#,$**%\"IG\"\"\"&%%betaG6#%\"mGF-%&DeltaGF-%\"tGF-!\"\"-F%6#,&F-F-*(
F,F-%\"rGF--%$sinG6#*(&%&sigmaG6#F1F-F2F-%\"xGF-F-F4" }{TEXT -1 4 "  =
 " }{XPPEDIT 18 0 "sqrt(1+r^2*sin(sigma[m]*Delta*x)^2);" "6#-%%sqrtG6#
,&\"\"\"F'*&%\"rG\"\"#-%$sinG6#*(&%&sigmaG6#%\"mGF'%&DeltaGF'%\"xGF'F*
F'" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 158 "This magnification factor is always larger than 1.0.  \+
Thus, any small perturbation in the solution will be amplified, and th
e scheme, although consistent, is " }{TEXT 281 15 "always unstable" }
{TEXT -1 186 ".  This means that any and all components of the solutio
n will grow without bound as time progresses.  Notice also that the re
sult shown above for the amplification factor, namely, that " }
{XPPEDIT 18 0 "abs(exp(-I*beta[m]*Delta*t))" "6#-%$absG6#-%$expG6#,$**
%\"IG\"\"\"&%%betaG6#%\"mGF,%&DeltaGF,%\"tGF,!\"\"" }{TEXT -1 15 " > 1
, requires " }{XPPEDIT 18 0 "beta[m];" "6#&%%betaG6#%\"mG" }{TEXT -1 
34 " to be an imaginary number, i.e., " }{XPPEDIT 18 0 "beta[m] = alph
a[m]+I*delta[m];" "6#/&%%betaG6#%\"mG,&&%&alphaG6#F'\"\"\"*&%\"IGF,&%&
deltaG6#F'F,F," }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "alpha[m] = Re(b
eta[m]);" "6#/&%&alphaG6#%\"mG-%#ReG6#&%%betaG6#F'" }{TEXT -1 5 " and \+
" }{XPPEDIT 18 0 "delta[m] = Im(beta[m]);" "6#/&%&deltaG6#%\"mG-%#ImG6
#&%%betaG6#F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 114 "In terms of the celerity of each compone
nt of the numerical solution we find that the celerity must be defined
 as " }{XPPEDIT 18 0 "alpha[m]/sigma[m];" "6#*&&%&alphaG6#%\"mG\"\"\"&
%&sigmaG6#F'!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Re(beta[m])/sigma
[m];" "6#*&-%#ReG6#&%%betaG6#%\"mG\"\"\"&%&sigmaG6#F*!\"\"" }{TEXT -1 
55 ".  To find an expression for this result, we start from" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 "     " }
{XPPEDIT 18 0 "exp(-I*beta[m]*Delta*t) = 1-I*r*sin(sigma[m]*Delta*x);
" "6#/-%$expG6#,$**%\"IG\"\"\"&%%betaG6#%\"mGF*%&DeltaGF*%\"tGF*!\"\",
&F*F**(F)F*%\"rGF*-%$sinG6#*(&%&sigmaG6#F.F*F/F*%\"xGF*F*F1" }{TEXT 
-1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 "" {TEXT 
-1 4 "    " }{XPPEDIT 18 0 "exp(-I*alpha[m]*Delta*t)*exp(delta[m]*Delt
a*t) = 1-I*r*sin(sigma[m]*Delta*x);" "6#/*&-%$expG6#,$**%\"IG\"\"\"&%&
alphaG6#%\"mGF+%&DeltaGF+%\"tGF+!\"\"F+-F&6#*(&%&deltaG6#F/F+F0F+F1F+F
+,&F+F+*(F*F+%\"rGF+-%$sinG6#*(&%&sigmaG6#F/F+F0F+%\"xGF+F+F2" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 "Using Euler's formula we can w
rite" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 4 "
    " }{XPPEDIT 18 0 "(cos(alpha[m]*Delta*t)-I*sin(alpha[m]*Delta*t))*
exp(delta[m]*Delta*t) = 1-I*r*sin(sigma[m]*Delta*x);" "6#/*&,&-%$cosG6
#*(&%&alphaG6#%\"mG\"\"\"%&DeltaGF.%\"tGF.F.*&%\"IGF.-%$sinG6#*(&F+6#F
-F.F/F.F0F.F.!\"\"F.-%$expG6#*(&%&deltaG6#F-F.F/F.F0F.F.,&F.F.*(F2F.%
\"rGF.-F46#*(&%&sigmaG6#F-F.F/F.%\"xGF.F.F9" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 7 "       " }}{PARA 0 "" 0 "" {TEXT -1 88 "Equating
 the real and imaginary parts of this equation we produce two equation
s, namely," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 
-1 5 "     " }{XPPEDIT 18 0 "sin(alpha[m]*Delta*t)*exp(delta[m]*Delta*
t);" "6#*&-%$sinG6#*(&%&alphaG6#%\"mG\"\"\"%&DeltaGF,%\"tGF,F,-%$expG6
#*(&%&deltaG6#F+F,F-F,F.F,F," }{TEXT -1 3 " = " }{XPPEDIT 18 0 "r*sin(
sigma[m]*Delta*x);" "6#*&%\"rG\"\"\"-%$sinG6#*(&%&sigmaG6#%\"mGF%%&Del
taGF%%\"xGF%F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}
{PARA 256 "" 0 "" {TEXT -1 4 "    " }{XPPEDIT 18 0 "cos(alpha[m]*Delta
*t)*exp(delta[m]*Delta*t) = 1;" "6#/*&-%$cosG6#*(&%&alphaG6#%\"mG\"\"
\"%&DeltaGF-%\"tGF-F--%$expG6#*(&%&deltaG6#F,F-F.F-F/F-F-F-" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
34 "Dividing term-by-term we find that" }}{PARA 256 "" 0 "" {XPPEDIT 
18 0 "tan(alpha[m]*Delta*t) = r*sin(sigma[m]*Delta*x);" "6#/-%$tanG6#*
(&%&alphaG6#%\"mG\"\"\"%&DeltaGF,%\"tGF,*&%\"rGF,-%$sinG6#*(&%&sigmaG6
#F+F,F-F,%\"xGF,F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 5 "Thus," }}{PARA 256 "" 0 "" {TEXT -1 3 "   \+
" }{XPPEDIT 18 0 "alpha[m] = atan(r*sin(sigma[m]*Delta*x))/(Delta*t);
" "6#/&%&alphaG6#%\"mG*&-%%atanG6#*&%\"rG\"\"\"-%$sinG6#*(&%&sigmaG6#F
'F.%&DeltaGF.%\"xGF.F.F.*&F6F.%\"tGF.!\"\"" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "and, the celerity of the " }
{TEXT 282 4 "m-th" }{TEXT -1 39 " component in the numerical solution \+
is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 
"a[m] = alpha[m]/sigma[m];" "6#/&%\"aG6#%\"mG*&&%&alphaG6#F'\"\"\"&%&s
igmaG6#F'!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "atan(r*sin(sigma[m]*
Delta*x))/(sigma[m]*Delta*t);" "6#*&-%%atanG6#*&%\"rG\"\"\"-%$sinG6#*(
&%&sigmaG6#%\"mGF)%&DeltaGF)%\"xGF)F)F)*(&F/6#F1F)F2F)%\"tGF)!\"\"" }
{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 45 "i.e., not a constant value but a function of " }{XPPEDIT 
18 0 "r;" "6#%\"rG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Delta*x/L[m] = si
gma[m]*Delta*x;" "6#/*(%&DeltaG\"\"\"%\"xGF&&%\"LG6#%\"mG!\"\"*(&%&sig
maG6#F+F&F%F&F'F&" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "Delta*t;" "6#*
&%&DeltaG\"\"\"%\"tGF%" }{TEXT -1 128 ".   Thus, every component of th
e numerical solution will propagate with different celerity not equal \+
to the convective celerity " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 
39 ".  This fact results in what is called " }{TEXT 283 20 "numerical \+
dispersion" }{TEXT -1 94 ", i.e., a phenomenon similar to physical dis
persion but caused by the numerical scheme itself." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Thus, the suggested numer
ical scheme is " }{TEXT 284 17 "always unstable, " }{TEXT -1 4 "and " 
}{TEXT 285 17 "always dispersive" }{TEXT -1 142 ", i.e., basically use
less for calculating a reliable numerical approximation to the partial
 differential equation of interest to our analysis." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 302 4 "NOTE" }{TEXT -1 329 ": \+
The procedure presented above for analyzind convergence and stability \+
of a numerical scheme, known as von Neumann (*) analysis, applies only
 to linear differential equations with periodic boundary conditions (t
o ensure applicability of the Fourier series components).  Thus, the a
nalysis would not apply to an equation such as " }{XPPEDIT 18 0 "diff(
u(x,t),t)+u(x,t)*diff(u(x,t),x) = 0;" "6#/,&-%%diffG6$-%\"uG6$%\"xG%\"
tGF,\"\"\"*&-F)6$F+F,F--F&6$-F)6$F+F,F+F-F-\"\"!" }{TEXT -1 293 " beca
use the second term in the equation is non-linear.   Linearizing the e
quation by some means can help in the application of von Neumann's ana
lysis for convergence and stability.  If a linearized scheme is not co
nvergent, most likely the original non-linear scheme will not converge
 either." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
3 "(*)" }{TEXT 303 16 "John von Neumann" }{TEXT -1 607 " (1903-1957), \+
a mathematician and chemical engineer, made significant contributions \+
to the sciences of quantum physics, mathematical logic, and meteorolog
y.  His contributions to computer science and game theory are also mon
umental.   He wrote about 150 papers on a number of subjects: physics,
 set theory, mathematical logic, topological groups, measure theory, e
rgodic theory, operator theory, continuous geometry, statistics, numer
ical analysis, shock waves, flow problems, hydrodynamics, aerodynamics
, ballistics, problems of detonation of explosives, meteorology, mathe
matical games and computer logic." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 3 "" 0 "" {TEXT -1 29 "Amplitude and phase portraits" }}{PARA 0 
"" 0 "" {TEXT -1 259 "The stability analysis shown earlier for a simpl
e linear partial differential equation and a specific numerical scheme
 produced two main results: (1) an amplification factor; and (2) celer
ity, for each component of the numerical solution.   We will define an
 " }{TEXT 286 23 "amplification parameter" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "R[1];" "6#&%\"RG6#\"\"\"" }{TEXT -1 150 " as the ratio of the ma
gnitudes of the numerical amplification factor to the true amplificati
on factor (which happens to be 1.0), i.e., for this case," }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "R[1];" "6#&%\"
RG6#\"\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "abs(exp(-I*beta[m]*Delta
*t))/1.0 = abs(1-I*r*sin(sigma[m]*Delta*x));" "6#/*&-%$absG6#-%$expG6#
,$**%\"IG\"\"\"&%%betaG6#%\"mGF.%&DeltaGF.%\"tGF.!\"\"F.-%&FloatG6$\"#
5F5F5-F&6#,&F.F.*(F-F.%\"rGF.-%$sinG6#*(&%&sigmaG6#F2F.F3F.%\"xGF.F.F5
" }{TEXT -1 4 "  = " }{XPPEDIT 18 0 "sqrt(1+r^2*sin(sigma[m]*Delta*x)^
2);" "6#-%%sqrtG6#,&\"\"\"F'*&%\"rG\"\"#-%$sinG6#*(&%&sigmaG6#%\"mGF'%
&DeltaGF'%\"xGF'F*F'" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "A pha
se parameter, " }{XPPEDIT 18 0 "R[2];" "6#&%\"RG6#\"\"#" }{TEXT -1 
120 ", is defined as the ration of the numerical celerity to that of t
he true (or analytical) celerity.  Thus, for this case," }}{PARA 256 "
" 0 "" {TEXT -1 4 "    " }{XPPEDIT 18 0 "R[2];" "6#&%\"RG6#\"\"#" }
{TEXT -1 3 " = " }{XPPEDIT 18 0 "a[m]/a;" "6#*&&%\"aG6#%\"mG\"\"\"F%!
\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "atan(r*sin(sigma[m]*Delta*x))/
(a*sigma[m]*Delta*t);" "6#*&-%%atanG6#*&%\"rG\"\"\"-%$sinG6#*(&%&sigma
G6#%\"mGF)%&DeltaGF)%\"xGF)F)F)**%\"aGF)&F/6#F1F)F2F)%\"tGF)!\"\"" }
{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 24 "Plots of the parameters " }{XPPEDIT 18 0 "R[1];" "6#&%\"R
G6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "R[2];" "6#&%\"RG6#\"\"#
" }{TEXT -1 36 " versus the dimensionless parameter " }{XPPEDIT 18 0 "
L[m]/(Delta*x);" "6#*&&%\"LG6#%\"mG\"\"\"*&%&DeltaGF(%\"xGF(!\"\"" }
{TEXT -1 3 " = " }{XPPEDIT 18 0 "2*Pi/(sigma[m]*Delta*x);" "6#*(\"\"#
\"\"\"%#PiGF%*(&%&sigmaG6#%\"mGF%%&DeltaGF%%\"xGF%!\"\"" }{TEXT -1 20 
" are referred to as " }{TEXT 287 18 "amplitude portrait" }{TEXT -1 5 
" and " }{TEXT 288 14 "phase portrait" }{TEXT -1 219 ", respectively. \+
  These \"portraits\" can be used to show graphically the stability, o
r lack thereof, of a numerical scheme.   Typically, the \"portraits\" \+
will show plots corresponding to different values of the parameter " }
{TEXT 289 1 "r" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "a*Delta*t/(Delta*x)
" "6#**%\"aG\"\"\"%&DeltaGF%%\"tGF%*&F&F%%\"xGF%!\"\"" }{TEXT -1 24 ".
   While the parameter " }{XPPEDIT 18 0 "L[m];" "6#&%\"LG6#%\"mG" }
{TEXT -1 50 " represents the characteristic wave length of the " }
{TEXT 292 4 "m-th" }{TEXT -1 101 " component of the numerical solution
, it can be taken to be the length of the solution domain, i.e., " }
{TEXT 293 7 "0 < x <" }{TEXT -1 1 " " }{XPPEDIT 18 0 "L[m];" "6#&%\"LG
6#%\"mG" }{TEXT -1 37 ".  Thus, the dimensionless parameter " }
{XPPEDIT 18 0 "L[s] = L[m]/(Delta*x);" "6#/&%\"LG6#%\"sG*&&F%6#%\"mG\"
\"\"*&%&DeltaGF,%\"xGF,!\"\"" }{TEXT -1 90 " relates the length of the
 solution domain to the grid size.  The smallest the grid size, " }
{XPPEDIT 18 0 "Delta*x;" "6#*&%&DeltaG\"\"\"%\"xGF%" }{TEXT -1 27 ", t
he largest the value of " }{XPPEDIT 18 0 "L[s];" "6#&%\"LG6#%\"sG" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 100 "Consider the amplitude and phase portraits of the scheme
 analyzed above.  First, the expression for " }{XPPEDIT 18 0 "R[1];" "
6#&%\"RG6#\"\"\"" }{TEXT -1 33 " can be written as a function of " }
{XPPEDIT 18 0 "L[s];" "6#&%\"LG6#%\"sG" }{TEXT -1 4 " =  " }{XPPEDIT 
18 0 "L[m]/(Delta*x)" "6#*&&%\"LG6#%\"mG\"\"\"*&%&DeltaGF(%\"xGF(!\"\"
" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "2*Pi/(sigma[m]*Delta*x)" "6#*(\"\"
#\"\"\"%#PiGF%*(&%&sigmaG6#%\"mGF%%&DeltaGF%%\"xGF%!\"\"" }{TEXT -1 
14 ", by writting " }{XPPEDIT 18 0 "sigma[m]*Delta*x = 2*Pi/L[s];" "6#
/*(&%&sigmaG6#%\"mG\"\"\"%&DeltaGF)%\"xGF)*(\"\"#F)%#PiGF)&%\"LG6#%\"s
G!\"\"" }{TEXT -1 61 ".   With this result, the amplification paramete
r is given by" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R[1]
;" "6#&%\"RG6#\"\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sqrt(1+r^2*sin
(2*Pi/L[s])^2);" "6#-%%sqrtG6#,&\"\"\"F'*&%\"rG\"\"#-%$sinG6#*(F*F'%#P
iGF'&%\"LG6#%\"sG!\"\"F*F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Also, the phase portrait can be
 plotted by using " }{XPPEDIT 18 0 "sigma[m]*Delta*x = 2*Pi/L[s];" "6#
/*(&%&sigmaG6#%\"mG\"\"\"%&DeltaGF)%\"xGF)*(\"\"#F)%#PiGF)&%\"LG6#%\"s
G!\"\"" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "a*Delta*t;" "6#*(%\"aG\"
\"\"%&DeltaGF%%\"tGF%" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "r*Delta*x;" "
6#*(%\"rG\"\"\"%&DeltaGF%%\"xGF%" }{TEXT -1 37 ", so that the phase pa
rameter becomes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{XPPEDIT 18 0 "R[2];" "6#&%\"RG6#\"\"#" }{TEXT -1 3 " = " }{XPPEDIT 
18 0 "atan(r*sin(2*Pi/L[s]))/(r*2*Pi/L[s]);" "6#*&-%%atanG6#*&%\"rG\"
\"\"-%$sinG6#*(\"\"#F)%#PiGF)&%\"LG6#%\"sG!\"\"F)F)**F(F)F.F)F/F)&F16#
F3F4F4" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "L[s]/(2*Pi*r);" "6#*&&%\"LG6
#%\"sG\"\"\"*(\"\"#F(%#PiGF(%\"rGF(!\"\"" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "atan(r*sin(2*Pi/L[s]));" "6#-%%atanG6#*&%\"rG\"\"\"-%$sinG6#*(\"
\"#F(%#PiGF(&%\"LG6#%\"sG!\"\"F(" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Thus, by letting " }
{XPPEDIT 18 0 "L[s];" "6#&%\"LG6#%\"sG" }{TEXT -1 18 " be between 0 an
d " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 65 ", we
 can produce the amplitude and phase portraits for values of " }
{XPPEDIT 18 0 "r;" "6#%\"rG" }{TEXT -1 87 " = 0.1, 0.5, 0.7, and 1.0. \+
  The plots are shown below.  First, the amplitude portrait:" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "res
tart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "R1:=(r,Ls) -> sqrt
(1+r^2*sin(2*Pi/Ls));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#R1Gf*6$%\"
rG%#LsG6\"6$%)operatorG%&arrowGF)-%%sqrtG6#,&\"\"\"F1*&)9$\"\"#F1-%$si
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[Z#>F-7$Fg\\l$\"3em`iT\"*\\g>F-7$F\\]l$\"3KA4DpSrv>F-7$$\"31+++-T9gYFf
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H/fCa[&F-7$Fe[p$!3cNu6&G'R<UF-7$Fggl$!3/bC/^Y4Q?F-7$$\"31+++;u3$*>Fhbl
$!37_IwL/\"GX$F07$F]\\p$\"3'fT;CY*[,9F-7$$\"3!)*****H#z)Q1#Fhbl$\"3'p`
#yx\\=pHF-7$F\\hl$\"3+BREV)**yD%F-7$Fe\\p$\"3U=q7H/xreF-7$Fahl$\"3KL\"
e&yW`+qF-7$Ffhl$\"3*yHMDd!y8()F-7$F[il$\"3C$QQ-O=)e(*F-7$F`il$\"3!oa>I
,')*o5Ffn7$Feil$\"3k19h%pGn9\"Ffn7$Fjil$\"3VJ0!prl7A\"Ffn7$F_jl$\"3qk(
z^q\"*eG\"Ffn7$Fdjl$\"3a[^v-mw_8Ffn7$Fijl$\"3V%p=\"oo&*>9Ffn7$F^[m$\"3
5GzgF1+x9Ffn7$Fc[m$\"3Sj\\HS:SP:Ffn7$Fh[m$\"33*)o\"))))4()f\"Ffn7$F]\\
m$\"3wRbF.xxd;Ffn7$Fb\\m$\"3(zFW\\1$=9<Ffn7$Fg\\m$\"3[ApB/#\\gx\"Ffn7$
F\\]m$\"3#3jmn&)>5$=Ffn7$Fa]m$\"3cnRd#>?\"*)=Ffn7$Ff]m$\"3_N\"yWSv7%>F
fn7$F[^m$\"3;U#[IP\"z(*>Ffn7$F`^m$\"3ETn,,E^]?Ffn7$Fe^m$\"3!)3l#o>q^5#
Ffn7$Fj^m$\"3Oz4w/#z\"e@Ffn7$F__m$\"3+.*3T1OK@#Ffn7$Fd_m$\"3a2(4!Q3&eE
#Ffn7$Fi_m$\"3qbn3-0D>BFfn7$F^`m$\"3'*zzi;9\"=P#Ffn7$Fc`m$\"3sw]!*)Ho(
>CFfn7$Fh`m$\"3s7(o,&yLuCFfn7$F]am$\"3lR@xeVzADFfn7$Fbam$\"3c-XMx$*4uD
Ffn7$Fgam$\"3U20#zshGi#Ffn7$F\\bm$\"37[yLOT*[n#Ffn-Fabm6&FcbmF(FdbmFdb
m-%+AXESLABELSG6%Q%L/Dx6\"Q#R2Fdbr-%%FONTG6#%(DEFAULTG-%*AXESSTYLEG6#%
$BOXG-%&TITLEG6#Q/Phase~PortraitFdbr-%%VIEWG6$;F($\"+3`=$G'!\"*Fibr" 
1 2 0 1 10 0 2 9 1 2 2 1.000000 47.000000 46.000000 0 0 "Curve 1" "Cur
ve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 144 "From the amplitude portraits shown above we noti
ce that the amplification parameter varies between 0.2 and 1.4 when th
e values of the parameter " }{TEXT 298 1 "r" }{TEXT -1 3 " = " }
{XPPEDIT 18 0 "a*Delta*t/(Delta*x)" "6#**%\"aG\"\"\"%&DeltaGF%%\"tGF%*
&F&F%%\"xGF%!\"\"" }{TEXT -1 32 " are smaller than 1.0 (i.e., if " }
{XPPEDIT 18 0 "a*Delta*t;" "6#*(%\"aG\"\"\"%&DeltaGF%%\"tGF%" }{TEXT 
-1 3 " < " }{XPPEDIT 18 0 "Delta*x;" "6#*&%&DeltaG\"\"\"%\"xGF%" }
{TEXT -1 62 "), the amplification parameter varies widely if the value
s of " }{XPPEDIT 18 0 "L[s] = L[m]/(Delta*x);" "6#/&%\"LG6#%\"sG*&&F%6
#%\"mG\"\"\"*&%&DeltaGF,%\"xGF,!\"\"" }{TEXT -1 15 " are less than " }
{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 47 ".   The a
mplification parameter peaks at about " }{XPPEDIT 18 0 "L[s];" "6#&%\"
LG6#%\"sG" }{TEXT -1 0 "" }{TEXT 299 4 " = 3" }{TEXT -1 24 ", and then
 decreases as " }{XPPEDIT 18 0 "L[s];" "6#&%\"LG6#%\"sG" }{TEXT -1 22 
" grows past the value " }{XPPEDIT 18 0 "L[s] = 3;" "6#/&%\"LG6#%\"sG
\"\"$" }{TEXT -1 18 ". As the value of " }{TEXT 300 1 "r" }{TEXT -1 
199 " grows larger than 1.0, the amplification parameter reaches large
r values.   Thus, the scheme proposed herein will produce relatively l
arge amplification parameters particularly for larger values of " }
{TEXT 301 1 "r" }{TEXT -1 25 " and for small values of " }{XPPEDIT 18 
0 "L[s];" "6#&%\"LG6#%\"sG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "The phase portrait shows the p
hase parameter as an oscillatory signal whose amplitude and wavelength
 increases with " }{XPPEDIT 18 0 "L[s];" "6#&%\"LG6#%\"sG" }{TEXT -1 
161 ", thus, confirming the observation pointed out earlier that the c
elerity of the numerical solution components is different and will pro
duce numerical dispersion." }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }}{PARA 
3 "" 0 "" {TEXT -1 48 "Creating amplitude and phase portraits in Matla
b" }}{PARA 0 "" 0 "" {TEXT -1 115 "The following script can be used to
 produce amplitude and phase portraits for the present finite-differen
ce scheme:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 72 "===============================================================
=========" }}{PARA 257 "" 0 "" {TEXT -1 2094 "% Script to plot amplitu
de and phase portraits for solving the\n% PDE:  diff(u(x,t),t)+ a*diff
(u(x,t),x) = 0, using forward\n% time derivative and centered space de
rivative.\n% The amplitude portrait consists of plotting the amplifica
tion\n% parameter R1 vs. Ls = L/Dx, for different values of r = a*Dt/D
x.\n% The phase portrait consists of plotting the phase parameter R2\n
% vs. Ls = L/Dx, for different values of r = a*Dt/Dx.\n\n% (1) First, \+
we produce the AMPLITUDE PORTRAIT for 0.1 < r < 1.0:\nLs = [0:0.1:100]
; r = [0.1,0.5,0.7,1.0]; cc = ['r','b','g','k','m'];\nn = length(Ls); \+
    m = length(r);\nR1 = zeros(n,m);\nfor j = 1:m\n   for i = 1:n \n  \+
    R1(i,j) = sqrt(1+r(j)^2*sin(2*pi/Ls(i)));\n   end;\nend;\nfigure(1
);hold;\nfor j = 1:m\n   plot(Ls,R1(:,j),cc(j));\nend;\nhold;\nxlabel(
'L/Dx');ylabel('R1');title('Amplitude Portrait, 0.1 < r < 1.0');\n\n% \+
(2) Next, we produce the AMPLITUDE PORTRAIT for 1.0 < r < 5.0:\nLs = [
0:0.1:100]; r = [1.0,1.5,2.0,5.0]; cc = ['r','b','g','k','m'];\nn = le
ngth(Ls);     m = length(r);\nR1 = zeros(n,m);\nfor j = 1:m\n   for i \+
= 1:n \n      R1(i,j) = sqrt(1+r(j)^2*sin(2*pi/Ls(i)));\n   end;\nend;
\nfigure(2);hold;\nfor j = 1:m\n   plot(Ls,R1(:,j),cc(j));\nend;\nhold
;\nxlabel('L/Dx');ylabel('R1');title('Amplitude Portrait, 1.0 < r < 5.
0');\n\n% (3) Next, we produce the PHASE PORTRAIT for 0.1 < r < 1.0:\n
Ls = [0:0.1:100]; r = [0.1,0.5,0.7,1.0]; cc = ['r','b','g','k','m'];\n
n = length(Ls);     m = length(r);\nR2 = zeros(n,m);\nfor j = 1:m\n   \+
for i = 1:n \n      R2(i,j) = (Ls(i)/(2*pi*r(j)))*atan(r(j)*sin(2*pi/L
s(i)));\n   end;\nend;\nfigure(3);hold;\nfor j = 1:m\n   plot(Ls,R2(:,
j),cc(j));\nend;\nhold;\nxlabel('L/Dx');ylabel('R2');title('Phase Port
rait, 0.1 < r < 1.0');\n\n% (4) Next, we produce the PHASE PORTRAIT fo
r 1.0 < r < 5.0:\nLs = [0:0.1:100]; r = [1.0,1.5,2.0,5.0]; cc = ['r','
b','g','k','m'];\nn = length(Ls);     m = length(r);\nR2 = zeros(n,m);
\nfor j = 1:m\n   for i = 1:n \n      R2(i,j) = (Ls(i)/(2*pi*r(j)))*at
an(r(j)*sin(2*pi/Ls(i)));\n   end;\nend;\nfigure(4);hold;\nfor j = 1:m
\n   plot(Ls,R2(:,j),cc(j));\nend;\nhold;\nxlabel('L/Dx');ylabel('R1')
;title('Phase Portrait, 1.0 < r < 5.0');" }}{PARA 0 "" 0 "" {TEXT -1 
71 "==================================================================
=====" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 11 "
Assignment " }}{PARA 0 "" 0 "" {TEXT -1 173 "NOTE:  This assignment is
 based on class notes for the class COMPUTATIONAL HYDRAULICS, as taugh
t by Dr. Forrest Holly in the Spring Semester 1985 at the University o
f Iowa. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
57 "Consider once more the linear advection equation, namely," }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "diff(
u(x,t),t)+a*diff(u(x,t),x) = 0;" "6#/,&-%%diffG6$-%\"uG6$%\"xG%\"tGF,
\"\"\"*&%\"aGF--F&6$-F)6$F+F,F+F-F-\"\"!" }{TEXT -1 1 "," }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "were " }{TEXT 304 
1 "a" }{TEXT -1 66 " is a constant.   Suppose that we want to solve th
is equation for " }{TEXT 305 6 "u(x,t)" }{TEXT -1 15 " in the domain \+
" }{TEXT 306 20 "0 < x < L, 0 < t < T" }{TEXT -1 25 " with boundary co
ndition " }{TEXT 307 9 "u(0,t) = " }{XPPEDIT 309 0 "u[1];" "6#&%\"uG6#
\"\"\"" }{TEXT 310 3 "(t)" }{TEXT -1 24 ", and initial condition " }
{TEXT 308 9 "u(x,0) = " }{XPPEDIT 311 0 "u[0];" "6#&%\"uG6#\"\"!" }
{TEXT 312 3 "(x)" }{TEXT -1 54 ".   The numerical solution will utiliz
e the so-called " }{TEXT 313 40 "upwind finite difference approximatio
ns " }{TEXT -1 28 "for the derivatives, namely," }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "diff(u(x,t),t) = (u[i
,j+1]-u[i,j])/(Delta*t);" "6#/-%%diffG6$-%\"uG6$%\"xG%\"tGF+*&,&&F(6$%
\"iG,&%\"jG\"\"\"F3F3F3&F(6$F0F2!\"\"F3*&%&DeltaGF3F+F3F6" }{TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 260 "" 0 "" {XPPEDIT 
18 0 "diff(u(x,t),x) = (u[i,j]-u[i-1,j])/(Delta*x);" "6#/-%%diffG6$-%
\"uG6$%\"xG%\"tGF**&,&&F(6$%\"iG%\"jG\"\"\"&F(6$,&F0F2F2!\"\"F1F6F2*&%
&DeltaGF2F*F2F6" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 251 "(a) Perform a stability analysis (von Ne
umann's analysis) on the upwind difference scheme that results from us
ing the finite difference approximations shown above in the linear adv
ection equation.  (Follow the approach shown earlier in this document)
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "(b) \+
Obtain expression for the amplification and phase parameters, " }
{XPPEDIT 18 0 "R[1](r,Ls);" "6#-&%\"RG6#\"\"\"6$%\"rG%#LsG" }{TEXT -1 
5 " and " }{XPPEDIT 18 0 "R[2](r,Ls);" "6#-&%\"RG6#\"\"#6$%\"rG%#LsG" 
}{TEXT -1 22 ", respectively, where " }{XPPEDIT 18 0 "L[s] = L/(Delta*
x);" "6#/&%\"LG6#%\"sG*&F%\"\"\"*&%&DeltaGF)%\"xGF)!\"\"" }{TEXT -1 5 
" and " }{XPPEDIT 18 0 "r = a*Delta*t/(Delta*x);" "6#/%\"rG**%\"aG\"\"
\"%&DeltaGF'%\"tGF'*&F(F'%\"xGF'!\"\"" }{TEXT -1 8 " (NOTE: " }
{XPPEDIT 18 0 "r;" "6#%\"rG" }{TEXT -1 49 " is sometimes referred to a
s the Courant number)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 73 "(c) Using MATLAB, plot amplitude and phase portraits
 for Courant numbers " }{TEXT 314 29 "r = 0.25, 0.5, 0.75, 1.0, 2.0" }
{TEXT -1 14 ", and for 0 < " }{XPPEDIT 18 0 "L/(Delta*x);" "6#*&%\"LG
\"\"\"*&%&DeltaGF%%\"xGF%!\"\"" }{TEXT -1 6 " < 30." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "(d) Based on the resul
ts shown in your amplitude and phase portraits discuss the stability o
f the upwind method." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{MARK "251 0" 48 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 1 1 
2 33 1 1 }
            
Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 17). StabilityNumericalSchemes.mws. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Civil_and_Environmental_Engineering/Numerical_Methods_in_Civil_Engineering/StabilityNumericalSchemes.mws-view.html. This work is licensed under a Creative Commons License Creative Commons License