📄 comparison of an implicite vs_ explicite one-dimensional finite differences program for solving the unsteady heat-flow-equation.mht

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Subject: comparison of an implicite vs. explicite one-dimensional finite differences program for solving the unsteady heat-flow-equation
Date: Sun, 12 Oct 2008 21:02:54 +0800
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<HTML><HEAD><TITLE>comparison of an implicite vs. explicite =
one-dimensional finite differences program for solving the unsteady =
heat-flow-equation</TITLE>
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<H1 align=3Dcenter><FONT face=3D"Helvetica, Arial, sans-serif"=20
color=3D#000000>comparison of an implicit vs. explicit one-dimensional =
finite=20
differences program for solving the unsteady heat-flow-equation =
</FONT></H1>
<P><FONT face=3D"Helvetica, Arial, sans-serif" color=3D#000000>The =
heat-flow=20
equation (without source-term):</FONT></P>
<P><FONT face=3D"Helvetica, Arial, sans-serif" color=3D#000000><IMG =
height=3D46=20
alt=3Dheatflow-equation hspace=3D8=20
src=3D"http://www.geomath.onlinehome.de/models/heatflow-equation.gif" =
width=3D117=20
align=3Dmiddle name=3Dheatflow-equation> </FONT></P>
<P></P>
<P><A =
href=3D"http://www.geomath.onlinehome.de/models/parameter.dat"><FONT=20
face=3D"Helvetica, Arial, sans-serif">The input data-file (identical for =
both=20
programs)</FONT></A></P>
<P><A href=3D"http://www.geomath.onlinehome.de/models/exphfeq.f"><FONT=20
face=3D"Helvetica, Arial, sans-serif">The explicit version (Fortran 77 =
source=20
code)</FONT></A></P>
<P><FONT face=3D"Helvetica, Arial, sans-serif"><A=20
href=3D"http://www.geomath.onlinehome.de/models/implhfeq.f">The implict =
version=20
(Fortran 77 source code)</A></FONT><A=20
href=3D"http://www.geomath.onlinehome.de/models/implhfeq.f"><FONT=20
face=3D"Helvetica, Arial, sans-serif"></FONT></A></P>
<P></P>
<P><FONT face=3D"Helvetica, Arial, sans-serif"><FONT=20
face=3D"Helvetica, Arial, sans-serif"><A=20
href=3D"http://www.geomath.onlinehome.de/models/compare.f">A small =
programm for=20
comparison of the results (Fortran 77 source =
code)<BR></A></FONT></FONT></P>
<P><FONT face=3D"Helvetica, Arial, sans-serif"><FONT=20
face=3D"Helvetica, Arial, sans-serif">The programs calculate the heat =
distribution=20
at a 100m long row in one year, where the row has the following thermal=20
properties:</FONT></FONT><FONT=20
face=3D"Helvetica, Arial, sans-serif"><BR></FONT></P>
<UL>
  <LI><FONT face=3D"Helvetica, Arial, sans-serif">Heat conduction =3D =
1.5 (W/(K=20
  m))</FONT>=20
  <LI><FONT face=3D"Helvetica, Arial, sans-serif">Density * specific =
heat =3D=20
  1,380,000 (J/(m=B3 M))</FONT>=20
  <LI><FONT face=3D"Helvetica, Arial, sans-serif">The time step is in =
both cases 1=20
  hour</FONT>=20
  <LI><FONT face=3D"Helvetica, Arial, sans-serif">Initial temperature =
=3D 283.15=20
  K</FONT>=20
  <LI><FONT face=3D"Helvetica, Arial, sans-serif">Spatial step size 1 =
m</FONT>=20
  <LI><FONT face=3D"Helvetica, Arial, sans-serif">At the first 3 grid =
nodes there=20
  is a Dirichlet border condition with a T of 283.15 + 50 (K)</FONT> =
</LI></UL><PRE></PRE>
<H1><FONT face=3D"Helvetica, Arial, sans-serif"><FONT=20
face=3D"Helvetica, Arial, sans-serif">The =
results:</FONT></FONT></H1><FONT=20
face=3D"Helvetica, Arial, sans-serif"></FONT>
<H2><FONT face=3D"Helvetica, Arial, sans-serif"><FONT=20
face=3D"Helvetica, Arial, sans-serif">Explicit</FONT></FONT></H2><FONT=20
face=3D"Helvetica, Arial, sans-serif"></FONT>
<P><FONT face=3D"Helvetica, Arial, sans-serif"><FONT=20
face=3D"Helvetica, Arial, sans-serif"><IMG height=3D600 alt=3D"explicit =
solution"=20
src=3D"http://www.geomath.onlinehome.de/models/EXPL.OUT.png" width=3D800 =
align=3Dleft=20
border=3D0 name=3DGrafik1> <BR clear=3Dleft></FONT></FONT></P><FONT=20
face=3D"Helvetica, Arial, sans-serif"></FONT>
<H2><FONT face=3D"Helvetica, Arial, sans-serif"><FONT=20
face=3D"Helvetica, Arial, sans-serif">Implicit</FONT></FONT></H2><FONT=20
face=3D"Helvetica, Arial, sans-serif"></FONT>
<P><FONT face=3D"Helvetica, Arial, sans-serif"><FONT=20
face=3D"Helvetica, Arial, sans-serif"><IMG height=3D600 alt=3D"implicite =
solution"=20
src=3D"http://www.geomath.onlinehome.de/models/IMPL.OUT.png" width=3D800 =
align=3Dleft=20
border=3D0 name=3DGrafik2> <BR clear=3Dleft></FONT></FONT></P><FONT=20
face=3D"Helvetica, Arial, sans-serif"></FONT>
<H2><FONT face=3D"Helvetica, Arial, sans-serif"><FONT=20
face=3D"Helvetica, Arial, sans-serif">The =
difference</FONT></FONT></H2><FONT=20
face=3D"Helvetica, Arial, sans-serif"></FONT>
<P><FONT face=3D"Helvetica, Arial, sans-serif"><FONT=20
face=3D"Helvetica, Arial, sans-serif">(really small - only at the =
borders there is=20
a bigger difference probably because of a malfunction of the=20
SY-Subroutine):</FONT></FONT></P><FONT=20
face=3D"Helvetica, Arial, sans-serif"></FONT>
<P><FONT face=3D"Helvetica, Arial, sans-serif"><FONT=20
face=3D"Helvetica, Arial, sans-serif"><IMG height=3D600 alt=3Dcomparison =

src=3D"http://www.geomath.onlinehome.de/models/DIFF.OUT.png" width=3D800 =
align=3Dleft=20
border=3D0 name=3DGrafik3> <BR clear=3Dleft></FONT></FONT></P><FONT=20
face=3D"Helvetica, Arial, sans-serif"><FONT=20
face=3D"Helvetica, Arial, sans-serif"><BR></FONT></FONT>
<HR width=3D"100%" SIZE=3D2>

<DIV align=3Dright>Author: <A =
href=3D"mailto:w.ruehaak@online.de">Wolfram=20
R=FChaak</A>, 28. May 2002<BR></DIV><BR><BR><BR></BODY></HTML>

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