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<html>
<head>
<title>
FD1D_HEAT_EXPLICIT - Time Dependent 1D Heat Equation, Finite Difference, Explicit Time Stepping
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FD1D_HEAT_EXPLICIT <br>
Finite Difference Solution of the<br>
Time Dependent 1D Heat Equation<br>
using Explicit Time Stepping
</h1>
<hr>
<p>
<b>FD1D_HEAT_EXPLICIT</b>
is a MATLAB library which
solves the time-dependent 1D heat equation, using the finite difference
method in space, and an explicit version of the method of lines to handle
integration in time.
</p>
<p>
This program solves
<pre>
dUdT - k * d2UdX2 = F(X,T)
</pre>
over the interval [A,B] with boundary conditions
<pre>
U(A,T) = UA(T),
U(B,T) = UB(T),
</pre>
over the time interval [T0,T1] with initial conditions
<pre>
U(X,T0) = U0(X)
</pre>
</p>
<p>
A second order finite difference is used to approximate the
second derivative in space.
</p>
<p>
The solver applies an
explicit forward Euler approximation to the first derivative in time.
</p>
<p>
The resulting finite difference form can be written as
<pre>
U(X,T+dt) - U(X,T) ( U(X-dx,T) - 2 U(X,T) + U(X+dx,T) )
------------------ = k * ------------------------------------ + F(X,T)
dt dx * dx
</pre>
or, assuming we have solved for all values of U at time T, we have
<pre>
U(X,T+dt) = U(X,T) + cfl * ( U(X-dx,T) - 2 U(X,T) + U(X+dx,T) ) + dt * F(X,T)
</pre>
where "cfl" is the Courant-Friedrichs-Loewy coefficient:
<pre>
cfl = k * dt / dx / dx
</pre>
In order for accurate results to be computed by this explicit method,
the cfl coefficient must be less than 0.5!
</p>
<p>
Other approaches would involve a fully implicit backward Euler approximation or
the Crank-Nicholson approximation. These latter two methods have
improved stability.
</p>
<p>
A second worthwhile change would be to replace the constant
heat conductivity <b>K</b> by a function <b>K(X,T)</b>. The
spatial variation would allow for the modeling of a region
divided into subregions of different materials.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<i>h_new</i> = <b>fd1d_heat_explicit</b> ( <i>x_num</i>, <i>x</i>, <i>t</i>, <i>dt</i>,
<i>cfl</i>, <i>@rhs</i>, <i>@bc</i>, <i>h</i> )
</blockquote>
where
<ul>
<li>
<i>x_num</i>, the number of equally spaced nodes;
</li>
<li>
<i>x</i>, the coordinates of the nodes;
</li>
<li>
<i>t</i>, the current time;
</li>
<li>
<i>dt</i>, the time step;
</li>
<li>
<i>cfl</i>, the Courant-Friedrichs-Loewy coefficient;
</li>
<li>
<i>@rhs</i>, a function to evaluate the right hand side source terms.
</li>
<li>
<i>@bc</i>, a function to enforce the boundary conditions.
</li>
<li>
<i>h</i>, the vector of temperatures at time T.
</li>
<li>
<i>h_new</i>, the vector of temperatures at time T+DT.
</li>
</ul>
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>FD1D_HEAT_EXPLICIT</b> is available in
<a href = "../../c_src/fd1d_heat_explicit/fd1d_heat_explicit.html">a C version</a> and
<a href = "../../cpp_src/fd1d_heat_explicit/fd1d_heat_explicit.html">a C++ version</a> and
<a href = "../../f77_src/fd1d_heat_explicit/fd1d_heat_explicit.html">a FORTRAN77 version</a> and
<a href = "../../f_src/fd1d_heat_explicit/fd1d_heat_explicit.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fd1d_heat_explicit/fd1d_heat_explicit.html">a MATLAB version</a> and
<a href = "../../py_src/fd1d_heat_explicit/fd1d_heat_explicit.html">a Python version</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/fd1d_advection_ftcs/fd1d_advection_ftcs.html">
FD1D_ADVECTION_FTCS</a>,
a MATLAB program which
applies the finite difference method to solve the time-dependent
advection equation ut = - c * ux in one spatial dimension, with
a constant velocity, using the FTCS method, forward time difference,
centered space difference.
</p>
<p>
<a href = "../../m_src/fd1d_burgers_lax/fd1d_burgers_lax.html">
FD1D_BURGERS_LAX</a>,
a MATLAB program which
applies the finite difference method and the Lax-Wendroff method
to solve the non-viscous time-dependent Burgers equation
in one spatial dimension.
</p>
<p>
<a href = "../../m_src/fd1d_burgers_leap/fd1d_burgers_leap.html">
FD1D_BURGERS_LEAP</a>,
a MATLAB program which
applies the finite difference method and the leapfrog approach
to solve the non-viscous time-dependent Burgers equation in one spatial dimension.
</p>
<p>
<a href = "../../m_src/fd1d_bvp/fd1d_bvp.html">
FD1D_BVP</a>,
a MATLAB program which
applies the finite difference method
to a two point boundary value problem in one spatial dimension.
</p>
<p>
<a href = "../../m_src/fd1d_heat_implicit/fd1d_heat_implicit.html">
FD1D_HEAT_IMPLICIT</a>,
a MATLAB program which
uses the finite difference method and implicit time stepping
to solve the time dependent heat equation in 1D.
</p>
<p>
<a href = "../../m_src/fd1d_heat_steady/fd1d_heat_steady.html">
FD1D_HEAT_STEADY</a>,
a MATLAB program which
uses the finite difference method to solve the steady (time independent)
heat equation in 1D.
</p>
<p>
<a href = "../../m_src/fd1d_predator_prey/fd1d_predator_prey.html">
FD1D_PREDATOR_PREY</a>,
a MATLAB program which
uses finite differences to solve a 1D predator prey problem.
</p>
<p>
<a href = "../../m_src/fd1d_wave/fd1d_wave.html">
FD1D_WAVE</a>,
a MATLAB program which
applies the finite difference method to solve the time-dependent
wave equation in one spatial dimension.
</p>
<p>
<a href = "../../m_src/fem1d/fem1d.html">
FEM1D</a>,
a MATLAB program which
applies the finite element
method, with piecewise linear basis functions, to a linear
two point boundary value problem;
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
George Lindfield, John Penny,<br>
Numerical Methods Using MATLAB,<br>
Second Edition,<br>
Prentice Hall, 1999,<br>
ISBN: 0-13-012641-1,<br>
LC: QA297.P45.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "fd1d_heat_explicit_cfl.m">fd1d_heat_explicit_cfl.m</a>,
computes the Courant-Friedrichs-Loewy coefficient;
</li>
<li>
<a href = "fd1d_heat_explicit.m">fd1d_heat_explicit.m</a>,
carries out the calculation;
</li>
<li>
<a href = "r8mat_write.m">r8mat_write.m</a>,
writes an R8MAT to a file.
</li>
<li>
<a href = "r8vec_write.m">r8vec_write.m</a>,
writes an R8VEC to a file.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>,
prints the YMDHMS date as a timestamp;
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "fd1d_heat_explicit_test.m">fd1d_heat_explicit_test.m</a>,
runs all the tests.
</li>
<li>
<a href = "fd1d_heat_explicit_test01.m">fd1d_heat_explicit_test01.m</a>,
runs test 1;
</li>
<li>
<a href = "fd1d_heat_explicit_test02.m">fd1d_heat_explicit_test02.m</a>,
runs test 2;
</li>
<li>
<a href = "fd1d_heat_explicit_test03.m">fd1d_heat_explicit_test03.m</a>,
runs test 3;
</li>
<li>
<a href = "fd1d_heat_explicit_test_output.txt">fd1d_heat_explicit_test_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>TEST01</b> runs with initial condition 50 everywhere, boundary conditions
of 90 on the left and 70 on the right, and no right hand side source term.
<ul>
<li>
<a href = "bc_test01.m">bc_test01.m</a>,
enforces the boundary conditions.
</li>
<li>
<a href = "h_test01.txt">h_test01.txt</a>,
the computed H data.
</li>
<li>
<a href = "ic_test01.m">ic_test01.m</a>,
enforces the initial condition.
</li>
<li>
<a href = "k_test01.m">k_test01.m</a>,
sets the conductivity.
</li>
<li>
<a href = "plot_test01.png">plot_test01.png</a>,
a PNG image of the solution.
</li>
<li>
<a href = "rhs_test01.m">rhs_test01.m</a>,
supplies the right hand side source term.
</li>
<li>
<a href = "t_test01.txt">t_test01.txt</a>,
the T data.
</li>
<li>
<a href = "x_test01.txt">x_test01.txt</a>,
the X data.
</li>
</ul>
</p>
<p>
<b>TEST02</b> uses an exact solution of g(x,t) = exp ( - t ) .* sin ( sqrt ( k ) * x ).
<ul>
<li>
<a href = "bc_test02.m">bc_test02.m</a>,
enforces the boundary conditions.
</li>
<li>
<a href = "exact_test02.m">exact_test02.m</a>,
evaluates the exact solution.
</li>
<li>
<a href = "g_test02.txt">g_test02.txt</a>,
the exact data.
</li>
<li>
<a href = "h_test02.txt">h_test02.txt</a>,
the computed H data.
</li>
<li>
<a href = "ic_test02.m">ic_test02.m</a>,
enforces the initial condition.
</li>
<li>
<a href = "k_test02.m">k_test02.m</a>,
sets the conductivity.
</li>
<li>
<a href = "plot_test02.png">plot_test02.png</a>,
a PNG image of the solution.
</li>
<li>
<a href = "plot_g_test02.png">plot_g_test02.png</a>,
a PNG image of the exact solution.
</li>
<li>
<a href = "rhs_test02.m">rhs_test02.m</a>,
supplies the right hand side source term.
</li>
<li>
<a href = "t_test02.txt">t_test02.txt</a>,
the T data.
</li>
<li>
<a href = "x_test02.txt">x_test02.txt</a>,
the X data.
</li>
</ul>
</p>
<p>
<b>TEST03</b> runs on the interval -5 <= X <= 5, with initial condition 15 on the
entire left and 25 on the entire right. The solution should settle down to a
straight line from the left boundary to the right.
<ul>
<li>
<a href = "bc_test03.m">bc_test03.m</a>,
enforces the boundary conditions.
</li>
<li>
<a href = "h_test03.txt">h_test03.txt</a>,
the computed H data.
</li>
<li>
<a href = "ic_test03.m">ic_test03.m</a>,
enforces the initial condition.
</li>
<li>
<a href = "k_test03.m">k_test03.m</a>,
sets the conductivity.
</li>
<li>
<a href = "plot_test03.png">plot_test03.png</a>,
a PNG image of the solution.
</li>
<li>
<a href = "rhs_test03.m">rhs_test03.m</a>,
supplies the right hand side source term.
</li>
<li>
<a href = "t_test03.txt">t_test03.txt</a>,
the T data.
</li>
<li>
<a href = "x_test03.txt">x_test03.txt</a>,
the X data.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 31 January 2012.
</i>
<!-- John Burkardt -->
</body>
</html>