FEM1D_HEAT_EXPLICIT
Finite Element Solution of the
Time Dependent 1D Heat Equation
using Explicit Time Stepping

FEM1D_HEAT_EXPLICIT is a Python library 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 program solves

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

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

        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 GNU LGPL license.

Languages:

FEM1D_HEAT_EXPLICIT is available in a MATLAB version and a Python version.

Related Data and Programs:

FD1D_HEAT_EXPLICIT, a Python library which uses the finite difference method and explicit time stepping to solve the time dependent heat equation in 1D.

FEM1D_BVP_LINEAR, a Python program 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:

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

Source Code:

• assemble_fem.py, assembles the finite element stiffness matrix and right hand side.
• assemble_mass.py, assembles the finite element mass matrix.
• basis_function.py, evaluates the linear basis functions.
• fem1d_heat_explicit.py, takes one explicit time step.
• fem1d_heat_explicit_cfl.py, computes the Courant-Friedrichs-Loewy coefficient;
• quadrature_set.py, sets the quadrature rule;
• r8mat_write.py, writes an R8MAT to a file.
• r8vec_write.py, writes an R8VEC to a file.
• reference_to_physical.py, maps reference data to a subinterval;
• timestamp.py, prints the YMDHMS date as a timestamp;

Examples and Tests:

• fem1d_heat_explicit_test.py, calls all the tests.
• fem1d_heat_explicit_test.sh, runs all the tests.
• fem1d_heat_explicit_test.txt, the output file.
• fem1d_heat_explicit_test01.py, runs test 1;
• fem1d_heat_explicit_test02.py, runs test 2;
• fem1d_heat_explicit_test03.py, runs test 3;
• fem1d_heat_explicit_test.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.

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