{VERSION 3 0 "IBM RISC UNIX" "3.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 
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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart; " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "u:=t->(sin(100/(t)));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"uGR6#%\"tG6\"6$%)operatorG%&arrowGF(-%$sinG6
#,$*&\"\"\"F19$!\"\"\"$+\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 270 "I0  := (u,t0,Delta)->Delta;\nI1  := (u,t0,Delta)->in
t(u(t), t=t0..t0+Delta);\nI00 := (u,t0,Delta)->Delta^2;\nI01 := (u,t0,
Delta)->int(u(t)*(t-t0), t=t0..t0+Delta);\nI10 := (u,t0,Delta)->I0(u,t
0,Delta)*I1(u,t0,Delta)-I01(u,t0,Delta);\nI11 := (u,t0,Delta)->I1(u,t0
,Delta)^2/2; " }{TEXT -1 77 "Verwendung des expliziten Doppelintegrals
 fuehrt zu sehr grossen Rechenzeiten" }{MPLTEXT 1 0 2 "\n\n" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#I0GR6%%\"uG%#t0G%&DeltaG6\"6$%)operatorG%
&arrowGF*9&F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I1GR6%%\"uG%#t
0G%&DeltaG6\"6$%)operatorG%&arrowGF*-%$intG6$-9$6#%\"tG/F4;9%,&F7\"\"
\"9&F9F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$I00GR6%%\"uG%#t0G%&
DeltaG6\"6$%)operatorG%&arrowGF**$)9&\"\"#\"\"\"F*F*F*" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$I01GR6%%\"uG%#t0G%&DeltaG6\"6$%)operatorG%&arro
wGF*-%$intG6$*&-9$6#%\"tG\"\"\",&F5F69%!\"\"F6/F5;F8,&F8F69&F6F*F*F*" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$I10GR6%%\"uG%#t0G%&DeltaG6\"6$%)o
peratorG%&arrowGF*,&*&-%#I0G6%9$9%9&\"\"\"-%#I1GF2F6F6-%$I01GF2!\"\"F*
F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$I11GR6%%\"uG%#t0G%&DeltaG6
\"6$%)operatorG%&arrowGF*,$*$)-%#I1G6%9$9%9&\"\"#\"\"\"#\"\"\"F7F*F*F*
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f:=[[x->x[2],x->0],[x->
-x[2], x->1]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG7$7$R6#%\"xG6
\"6$%)operatorG%&arrowGF*&9$6#\"\"#F*F*F*\"\"!7$RF(F*F+F*,$F.!\"\"F*F*
F*\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots);" }
}{PARA 12 "" 1 "" {XPPMATH 20 "6#7U%(animateG%*animate3dG%-animatecurv
eG%-changecoordsG%,complexplotG%.complexplot3dG%*conformalG%,contourpl
otG%.contourplot3dG%*coordplotG%,coordplot3dG%-cylinderplotG%,densityp
lotG%(displayG%*display3dG%*fieldplotG%,fieldplot3dG%)gradplotG%+gradp
lot3dG%-implicitplotG%/implicitplot3dG%(inequalG%-listcontplotG%/listc
ontplot3dG%0listdensityplotG%)listplotG%+listplot3dG%+loglogplotG%(log
plotG%+matrixplotG%(odeplotG%'paretoG%*pointplotG%,pointplot3dG%*polar
plotG%,polygonplotG%.polygonplot3dG%4polyhedra_supportedG%.polyhedrapl
otG%'replotG%*rootlocusG%,semilogplotG%+setoptionsG%-setoptions3dG%+sp
acecurveG%1sparsematrixplotG%+sphereplotG%)surfdataG%)textplotG%+textp
lot3dG%)tubeplotG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "heun_
ct:= proc(N,show)\n \nlocal n, Delta, x, y, xf, yf, u1, u01, u11, A, B
, xe, ye, C, erreuler, errheun:\n\nDelta := evalf(1/N);\n\n\n" }{TEXT 
-1 38 "Heun: x,y \nEuler: xf, yf\nExakt: xe, ye" }{MPLTEXT 1 0 1644 "
\n\nx[0]:=0: y[0]:=0;\nxf[0]:=0: yf[0]:=0;\nxe[0]:=0: ye[0]:=0;\nerrhe
un:=0: erreuler:=0:\n\nfor n from 0 to N do\n  u1  := I1(u, Delta*n, D
elta):\n  u01 := I01(u, Delta*n, Delta):\n  u11 := I11(u, Delta*n, Del
ta):\n\n  x[n+1]:=evalf(x[n] + Delta*f[1][1]([x[n], y[n]]) +  u1*f[2][
1]([x[n], y[n]])\n                     + (u01-u11)                    \+
                 ):\n  y[n+1]:=evalf(y[n] + Delta*f[1][2]([x[n], y[n]]
) +  u1*f[2][2]([x[n], y[n]])):\n   \n  xf[n+1]:=evalf(xf[n] + Delta*f
[1][1]([xf[n], yf[n]]) +  u1*f[2][1]([xf[n], yf[n]])):\n  yf[n+1]:=eva
lf(yf[n] + Delta*f[1][2]([xf[n], yf[n]]) +  u1*f[2][2]([xf[n], yf[n]])
):\n\n  xe[n+1]:=evalf(I10(u,0,Delta*(n+1))-I11(u,0,Delta*(n+1))):\n  \+
ye[n+1]:=evalf(I1(u,0,Delta*(n+1))):\n\n  errheun  := max(errheun, abs
(xe[n+1]-x[n+1])):\n  erreuler := max(erreuler,abs(xe[n+1]-xf[n+1])):
\n\nod:\n\nprint(x[N], y[N], xf[N], yf[N], xe[N], ye[N]):\nprint(erreu
ler, errheun, erreuler*N, errheun*N^2):\n\nif (show=1) then\nA:=PLOT(C
URVES([seq([evalf(i*Delta),x[i]],i=0..N)]),COLOR(RGB,0,0,0),THICKNESS(
1),LINESTYLE(2),TITLE(\"\"),FONT(HELVETICA,OBLIQUE,10)):\n     B:=PLOT
(CURVES([seq([evalf(i*Delta),xe[i]],i=0..N)]),COLOR(RGB,0,0,0),THICKNE
SS(1),LINESTYLE(1),FONT(HELVETICA,OBLIQUE,10)):\n\nC:=PLOT(CURVES([seq
([evalf(i*Delta),xf[i]],i=0..N)]),COLOR(RGB,0,0,0),THICKNESS(1),LINEST
YLE(3),FONT(HELVETICA,OBLIQUE,10)):\nprint(display([A,B,C]));\n \n\n\n
#A:=PLOT(CURVES([seq([evalf(i*Delta),y[i]],i=0..N)]),COLOR(HUE,0.9),TH
ICKNESS(1),TITLE(\"y(t): Exakt vs. Heun\"),FONT(HELVETICA,OBLIQUE,10))
:\n \n#B:=PLOT(CURVES([seq([evalf(i*Delta),ye[i]],i=0..N)]),COLOR(HUE,
0.7),THICKNESS(1),FONT(HELVETICA,OBLIQUE,10)):\n#print(display([A,B]))
;\nfi:\nend:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "heun_ct(1
0,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"+brC9p!#8$\"*?to^)!#6$\"+z
?Bun!#9F&$\"+XNYg>F+$\")K(o^)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$
\"+6%)y*>\"!#7$\"+,3?=n!#8$F$!#6$F'F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "5 0 2" 1019 }{VIEWOPTS 1 1 0 1 1 1803 }