FD1D_HEAT_EXPLICIT
Finite Difference Solution of the
Time Dependent 1D Heat Equation
using Explicit Time Stepping

FD1D_HEAT_EXPLICIT is a C++ 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.

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

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

Other approaches would involve a fully implicit backward Euler approximation or the Crank-Nicholson approximation. These latter two methods have improved stability.

A second worthwhile change would be to replace the constant heat conductivity K by a function K(X,T). The spatial variation would allow for the modeling of a region divided into subregions of different materials.

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_EXPLICIT is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a Python version

Related Data and Programs:

FD1D_BURGERS_LAX, a C++ 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.

FD1D_BURGERS_LEAP, a C++ program which applies the finite difference method and the leapfrog approach to solve the non-viscous time-dependent Burgers equation in one spatial dimension.

FD1D_BVP, a C++ program which applies the finite difference method to a two point boundary value problem in one spatial dimension.

FD1D_HEAT_IMPLICIT, a C++ program which uses the finite difference method and implicit time stepping to solve the time dependent heat equation in 1D.

FD1D_HEAT_STEADY, a C++ program which uses the finite difference method to solve the steady (time independent) heat equation in 1D.

FD1D_PREDATOR_PREY, a C++ program which uses finite differences to solve a 1D predator prey problem.

FD1D_WAVE, a C++ program which applies the finite difference method to solve the time-dependent wave equation utt = c * uxx in one spatial dimension.

FEM1D, a C++ program which applies the finite element method, with piecewise linear basis functions, to a linear two point boundary value problem;

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_explicit.cpp, the source code.
• fd1d_heat_explicit.hpp, the include file.
• fd1d_heat_explicit.sh, commands to compile the source code.

Examples and Tests:

• fd1d_heat_explicit_prb.cpp, a sample calling program.
• fd1d_heat_explicit_prb.sh, BASH commands to compile and run the sample program.
• fd1d_heat_explicit_prb_output.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.

• plot_test01.png, a PNG image of the solution, using the MATLAB MESH command to emphasize the method of lines approach underlying the solution.
• h_test01.txt, the computed H data.
• t_test01.txt, the T data.
• x_test01.txt, the X data.

List of Routines:

• FD1D_HEAT_EXPLICIT: Finite difference solution of 1D heat equation.
• FD1D_HEAT_EXPLICIT_CFL: compute the Courant-Friedrichs-Loewy coefficient.
• R8MAT_WRITE writes an R8MAT file.
• R8VEC_LINSPACE_NEW creates a vector of linearly spaced values.
• R8VEC_WRITE writes an R8VEC file.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.