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{SECT 0 {PARA 18 "" 0 "" {TEXT -1 45 "Euler and Backward Euler Time Di
scretizations" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 157 "This worksheet illustrates the Euler and Backward Euler \+
 methods for time discretization for a simple one-dimensional unsteady
 convection diffusion equation." }}{PARA 0 "" 0 "" {TEXT -1 33 "Restar
t Maple and load libraries." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "with(LinearA
lgebra):with(plots):" }}}{PARA 0 "" 0 "" {TEXT -1 24 "We consider the \+
equation" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "diff(u,t)-1/5*diff(u,x,x)
+4/5*diff(u,x) = 0;" "6#/,(-%%diffG6$%\"uG%\"tG\"\"\"*(F*F*\"\"&!\"\"-
F&6%F(%\"xGF0F*F-*(\"\"%F*F,F--F&6$F(F0F*F*\"\"!" }{TEXT -1 6 "      \+
" }{XPPEDIT 18 0 "u(0,t) = 0;" "6#/-%\"uG6$\"\"!%\"tGF'" }{TEXT -1 5 "
     " }{XPPEDIT 18 0 "u(Pi,t) = 0;" "6#/-%\"uG6$%#PiG%\"tG\"\"!" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "with initial condition" }
}{PARA 257 "" 0 "" {XPPEDIT 18 0 "u(x,0) = exp(2*x)*sin(x);" "6#/-%\"u
G6$%\"xG\"\"!*&-%$expG6#*&\"\"#\"\"\"F'F/F/-%$sinG6#F'F/" }{TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 39 "the exact solution for this equation
 is" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "u(x,t) = exp(2*x-t)*sin(x);" "
6#/-%\"uG6$%\"xG%\"tG*&-%$expG6#,&*&\"\"#\"\"\"F'F0F0F(!\"\"F0-%$sinG6
#F'F0" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "Set the parameter
s... things (notation) should be similar to previous Maple worksheet" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "k:=4/5;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"kG#\"\"%\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "nu:=1/5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#nuG#\"
\"\"\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 26 "the number of grid points \+
" }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 22 " and the mesh spacing " 
}{XPPEDIT 18 0 "h;" "6#%\"hG" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 6 "n:=41;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#T
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "h:=Pi/(n-1);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"hG,$%#PiG#\"\"\"\"#S" }}}{PARA 0 "" 0 "
" {TEXT -1 15 "the time step k" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 13 "deltaT:=0.05;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'deltaTG$\"\"
&!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 77 "construct the matrix and vector
s (unknowns and r.h.s.) as in previous example" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 15 "A:=Matrix(n,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"AG-%'RTABLEG6+\")k$yf\"%)anythingG%'MatrixG%,rectangularG%.Fort
ran_orderG7\"\"\"#;\"\"\"\"#TF/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 13 "f:=Vector(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG-%'RTAB
LEG6*\")#*GB=%)anythingG&%'VectorG6#%'columnG%,rectangularG%.Fortran_o
rderG7\"\"\"\";F1\"#T" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "f[1
]:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"fG6#\"\"\"\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "f[n]:=0;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"fG6#\"#T\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "for i from 2 to n-1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 24 "A[i,i]:=2*nu/h^2+k/(2*h)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "
end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for i from 2 to \+
n-1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "A[i,i-1]:=-k/(2*h)-nu/h^2
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 22 "for i from 2 to n-1 do" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "A[i,i+1]:=-nu/h^2" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
7 "end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Id:=IdentityM
atrix(n,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#IdG-%'RTABLEG6+\")We
<;%)anythingG%'MatrixG%&emptyG%.Fortran_orderG7#%)identityG\"\"#;\"\"
\"\"#TF0" }}}{PARA 0 "" 0 "" {TEXT -1 11 "The matrix " }{XPPEDIT 18 0 
"A;" "6#%\"AG" }{TEXT -1 71 " above is the matrix obtained from semi-d
iscretization, and the matrix " }{XPPEDIT 18 0 "Ap;" "6#%#ApG" }{TEXT 
-1 47 " below is the one for the fully discrete system" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Ap:=MatrixAdd(Id,A,1,-deltaT);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ApG-%'RTABLEG6+\")CoF=%)anythingG%'
MatrixG%,rectangularG%.Fortran_orderG7\"\"\"#;\"\"\"\"#TF/" }}}{PARA 
0 "" 0 "" {TEXT -1 22 "set initial conditions" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 14 "u0:=Vector(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#u0G-%'RTABLEG6*\")#4L#=%)anythingG&%'VectorG6#%'columnG%,rectangu
larG%.Fortran_orderG7\"\"\"\";F1\"#T" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "for i from 2 to n-1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 34 "u0[i]:=exp(2*(i-1)*h)*sin((i-1)*h)" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "end do:" }}}{PARA 0 "" 0 "" {TEXT -1 70 "The following
 are 5 time steps for the explicit method (Forward Euler)" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "u1:=MatrixVectorMultiply(Ap,u0);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u1G-%'RTABLEG6*\")sJB=%)anythingG&
%'VectorG6#%'columnG%,rectangularG%.Fortran_orderG7\"\"\"\";F1\"#T" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "u2:=MatrixVectorMultiply(Ap
,u1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u2G-%'RTABLEG6*\")#HL#=%)a
nythingG&%'VectorG6#%'columnG%,rectangularG%.Fortran_orderG7\"\"\"\";F
1\"#T" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "u3:=MatrixVectorMu
ltiply(Ap,u2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u3G-%'RTABLEG6*\"
)sLB=%)anythingG&%'VectorG6#%'columnG%,rectangularG%.Fortran_orderG7\"
\"\"\";F1\"#T" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "u4:=Matrix
VectorMultiply(Ap,u3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u4G-%'RTA
BLEG6*\")#\\L#=%)anythingG&%'VectorG6#%'columnG%,rectangularG%.Fortran
_orderG7\"\"\"\";F1\"#T" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "
u5:=MatrixVectorMultiply(Ap,u4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#u5G-%'RTABLEG6*\")7OB=%)anythingG&%'VectorG6#%'columnG%,rectangularG%
.Fortran_orderG7\"\"\"\";F1\"#T" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT -1 74 "set up vectors for graphics to display the exact and appr
oximate solutions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "q0:=[se
q([(i-1)*h,u0[i]],i=1..n)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
34 "q1:=[seq([(i-1)*h,u1[i]],i=1..n)]:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "q2:=[seq([(i-1)*h,u2[i]],i=1..n)]:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 34 "q3:=[seq([(i-1)*h,u3[i]],i=1..n)]:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "q4:=[seq([(i-1)*h,u4[i]],i=1
..n)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "q5:=[seq([(i-1)*h
,u5[i]],i=1..n)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "pq0:=l
istplot(q0,thickness=2,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "ex0:=plot(exp(2*x)*sin(x),x=0..Pi):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 42 "pq1:=listplot(q1,thickness=2,color=black)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ex1:=plot(exp(2*x-delt
aT)*sin(x),x=0..Pi):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "pq2
:=listplot(q2,thickness=2,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "ex2:=plot(exp(2*x-2*deltaT)*sin(x),x=0..Pi):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "pq3:=listplot(q3,thickness=2
,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "ex3:=plot
(exp(2*x-3*deltaT)*sin(x),x=0..Pi):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "pq4:=listplot(q4,thickness=2,color=black):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "ex4:=plot(exp(2*x-4*deltaT)*
sin(x),x=0..Pi):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "pq5:=li
stplot(q5,thickness=2,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "ex5:=plot(exp(2*x-5*deltaT)*sin(x),x=0..Pi):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display(pq0,ex0);" }}{PARA 
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M7R0
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