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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 33 "The One Dimensional Heat
 Equation" }{TEXT 259 0 "" }{TEXT 260 23 " - The Leap-Frog Scheme" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "restart; with(plots): with(L
inearAlgebra):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name chang
ecoords has been redefined\n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 "
Analytic Solution " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "If u(x,t) is
 the temperature at position x and time t the one dimensional " }
{TEXT 261 13 "heat equation" }{TEXT -1 13 " is given by:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "heat := diff(u(x,t),t)=diff(u(x,t),
x,x);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%%heatG/-%%diffG6$-%\"uG6$%\"xG%\"tGF--F'6$F)-%\"$G6$F,\"\"#" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "In the lectures, we have found the
 particular solution " }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "u_part := (x,t) -> exp( -(k*Pi)^2 *t) * sin(k*Pi*x);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'u_partGR6$%\"xG%\"tG6\"6$%)oper
atorG%&arrowGF)*&-%$expG6#,$*()%\"kG\"\"#\"\"\")%#PiGF5F69%F6!\"\"F6-%
$sinG6#*(F4F6F8F69$F6F6F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "
If k is an integer," }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 7 "k := 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG\"\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "we can see that u_part satifies th
e heat equation" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
30 "subs(u(x,t)=u_part(x,t),heat);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$*&-%$ex
pG6#,$*&)%#PiG\"\"#\"\"\"%\"tGF0!\"\"F0-%$sinG6#*&F.F0%\"xGF0F0F1-F%6$
F'-%\"$G6$F7F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*()%#PiG\"\"#\"\"
\"-%$expG6#,$*&F&F)%\"tGF)!\"\"F)-%$sinG6#*&F'F)%\"xGF)F)F0F$" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "and the following boundary conditi
ons:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "u_0 := u
_part(0,t); u_1 := u_part(1,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$
u_0G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$u_1G\"\"!" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 23 "and initial conditions:" }{MPLTEXT 1 0 0 
"" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "u_part(x,0);" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 41 "u_init := unapply( eval(u_part(x,0)), x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#*&%#PiG\"\"\"%\"xGF(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'u_initGR6#%\"xG6\"6$%)operatorG%&ar
rowGF(-%$sinG6#*&%#PiG\"\"\"9$F1F(F(F(" }}}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 46 "Second Order Implicit Scheme (Crank Nicholson)" }}{PARA 
0 "" 0 "" {TEXT -1 217 "The leap frog scheme requires numerical soluti
ons for at least two time steps in order to start.\nWe will use the im
plicit second order Crank-Nicholson scheme to compute the numerical so
lution for the first time step. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 73 "n := 20; h := 1/n;\ntau := 1/200; steps := 10;\nv := array(0..
n,0..steps);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#?" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"hG#\"\"\"\"#?" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$tauG#\"\"\"\"$+#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%&stepsG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%&arrayG6%;\"
\"!\"#?;F)\"#57\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "We initializ
e our numerical solution:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "for j \+
from 1 to n-1 do \n   v[j,0] := u_init(j*h)\nend do:" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 32 "and set the boundary conditions:" }{MPLTEXT 1 
0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "for m from 0 to steps do\n
   v[0,m] := 0;\n   v[n,m] := 0;\nend do:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 69 "The iteration matrix is similar to that of the simple imp
licit scheme" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 116 
"tridiag := (i,j) -> `if`(i=j,2,\n                             `if`(i=
j+1 or i+1=j, -1,0)); \nA := Matrix(n-1,tridiag);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(tridiagGR6$%\"iG%\"jG6\"6$%)operatorG%&arrowGF)-%#if
G6%/9$9%\"\"#-F.6%5/F1,&F2\"\"\"F9F9/,&F1F9F9F9F2!\"\"\"\"!F)F)F)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6+\"*!eG$Q\"%)anything
G%'MatrixG%,rectangularG%.Fortran_orderG7\"\"\"#;\"\"\"\"#>F/" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "However, we require an additional \+
weighting parameter " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 1 
":" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "M := Matri
x(n-1,shape=identity) + (1/2)*tau/h^2*A;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"MG-%'RTABLEG6+\"*7g.P\"%)anythingG%'MatrixG%,rectangularG%.F
ortran_orderG7\"\"\"#;\"\"\"\"#>F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 88 "Compute the solution for the first time step (computing the rig
ht hand side in b first):" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 163 "b := < seq( v[j,0] + (1/2)*tau/h^2*( v[j-1,0] - 2*v[
j,0] + v[j+1,0]) ,j=1..n-1) >;\ntmp := LinearSolve(M, b);\nfor j from \+
1 to n-1 do\n   v[j,1] := tmp[j]:   \nend do:" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"bG-%'RTABLEG6*\"*!ozp8%)anythingG&%'VectorG6#%'colu
mnG%,rectangularG%.Fortran_orderG7\"\"\"\";F1\"#>" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$tmpG-%'RTABLEG6*\"*/-RP\"%)anythingG&%'VectorG6#%'co
lumnG%,rectangularG%.Fortran_orderG7\"\"\"\";F1\"#>" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 50 "And plot the results for the first two time ste
ps:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "plot1 :=
 listplot( [seq( v[j,0], j=0..n)]):\nplot2 := listplot( [seq( v[j,1], \+
j=0..n)]):\ndisplay(plot1,plot2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 
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{PARA 0 "" 0 "" {TEXT -1 67 "Now we compute the remaining time steps u
sing the leap-frog scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
169 "for m from 1 to steps-1 do\n   for j from 1 to n-1 do\n      v[j,
m+1] := v[j,m-1] + 2*tau*(\n                    v[j-1,m] - 2*v[j,m] + \+
v[j+1,m] )/h^2:   \n   end do:\nend do:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 128 "Finally, we plot the results. The size of the time step \+
has to bo chosen very carefully. The scheme is likely to explode . . .
  " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "resplot := [seq( listplot( [s
eq( v[j,m], j=0..n)] ), m=1..8)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
33 "display(resplot,insequence=true);" }}{PARA 13 "" 1 "" {GLPLOT2D 
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