FD1D_HEAT_IMPLICIT
Finite Difference Solution of the
Time Dependent 1D Heat Equation
using Implicit Time Stepping

FD1D_HEAT_IMPLICIT is a Python program which solves the time-dependent 1D heat equation, using the finite difference method in space, and an implicit 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)
      

A second order finite difference is used to approximate the second derivative in space.

The solver applies an implicit backward Euler approximation to the first derivative in time.

The resulting finite difference form can be written as

       U(X,T+dt) - U(X,T)                     ( U(X-dx,+dtT) - 2 U(X,+dtT) + U(X+dx,+dtT) )
       ------------------  = F(X,T+dt) + k *  ---------------------------------------------
                dt                                   dx * dx
      

            -     k * dt / dx / dx   * U(X-dt,T+dt)
      + ( 1 + 2 * k * dt / dx / dx ) * U(X,   T+dt)
            -     k * dt / dx / dx   * U(X+dt,T+dt)
      =               dt             * F(X,   T+dt)
      +                                U(X,   T)
      

Licensing:

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

Languages:

FD1D_HEAT_IMPLICIT is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

FD1D_ADVECTION_LAX_WENDROFF, a Python program which applies the finite difference method (FDM) to solve the time-dependent advection equation ut = - c * ux in one spatial dimension, with a constant velocity, using the Lax-Wendroff method to treat the time derivative.

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

FD2D_HEAT_STEADY, a Python program which uses the finite difference method (FDM) to solve the steady (time independent) heat equation in 2D.

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:

• fd1d_heat_implicit.m, carries out the calculation;
• fd1d_heat_implicit_cfl.py, computes the Courant-Friedrichs-Loewy coefficient;
• fd1d_heat_implicit_matrix.py, sets the system matrix.
• r8mat_write.py, writes an R8MAT to a file.
• r8vec_write.py, writes an R8VEC to a file.
• timestamp.py, prints the YMDHMS date as a timestamp;

Examples and Tests:

• fd1d_heat_implicit_test.py, calls all the tests.
• fd1d_heat_implicit_test.sh, runs all the tests.
• fd1d_heat_implicit_test01.py, runs test 1;
• fd1d_heat_implicit_test02.py, runs test 2;
• fd1d_heat_implicit_test03.py, runs test 3;
• fd1d_heat_implicit_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 H(X,T) 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_test031.png, a PNG image of the solution.
• t_test03.txt, the T data.
• x_test03.txt, the X data.

You can go up one level to the Python source codes.