fem1d_heat_explicit

fem1d_heat_explicit, a Python code which solves the time-dependent 1D heat equation, using the finite element method in space, and an explicit version of the method of lines to handle integration in time.

This code solves

        dUdT - k * d2UdX2 = F(X,T)

over the interval [A,B] with boundary conditions

        U(A,T) = UA(T),
        U(B,T) = UB(T),

over the time interval [T0,T1] with initial conditions

        U(X,T0) = U0(X)

The spatial derivatives are approximated using the finite element method, with piecewise linear elements.

The solver applies an explicit forward Euler approximation to the first derivative in time.

Licensing:

The computer code and data files described and made available on this web page are distributed under the MIT license

Languages:

fem1d_heat_explicit is available in a MATLAB version and a Python version.

Related Data and codes:

fd1d_heat_explicit, a Python code which uses the finite difference method and explicit time stepping to solve the time dependent heat equation in 1D.

fem1d_bvp_linear, a Python code which applies the finite element method (FEM), with piecewise linear elements, to a two point boundary value problem (BVP) in one spatial dimension, and compares the computed and exact solutions with the L2 and seminorm errors.

Reference:

George Lindfield, John Penny,
Numerical Methods Using MATLAB,
Second Edition,
Prentice Hall, 1999,
ISBN: 0-13-012641-1,
LC: QA297.P45.

Source Code:

fem1d_heat_explicit.py, the source code.
fem1d_heat_explicit.sh, runs all the tests.
fem1d_heat_explicit.txt, the output file.

TEST01 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.

h_test01.txt, the computed H data.
plot_test01.png, a PNG image of the solution.
t_test01.txt, the T data.
x_test01.txt, the X data.

TEST02 uses an exact solution of g(x,t) = exp ( - t ) .* sin ( sqrt ( k ) * x ).

g_test02.txt, the exact data.
h_test02.txt, the computed H data.
plot_test02.png, a PNG image of the solution.
t_test02.txt, the T data.
x_test02.txt, the X data.

TEST03 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.

h_test03.txt, the computed H data.
plot_test03.png, a PNG image of the solution.
t_test03.txt, the T data.
x_test03.txt, the X data.

Last revised on 07 November 2014.