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

FD1D_HEAT_EXPLICIT is a C program 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.

Related Data and Programs:

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.

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

FEM_50_HEAT is MATLAB program which applies the finite element method to solve the 2D heat equation.

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

FEM1D_ADAPTIVE is a C++ program which applies the finite element method to a linear two point boundary value problem in a 1D region, using adaptive refinement to improve the solution.

FEM1D_NONLINEAR is a C++ program which applies the finite element method to a nonlinear two point boundary value problem in a 1D region.

FEM1D_PMETHOD is a C++ program which applies the p-method version of the finite element method to a linear two point boundary value problem in a 1D region.

FEM2D_HEAT is a C++ program which applies the finite element method to solve the 2D heat equation.

FREE_FEM_HEAT is a C++ program which applies the finite element method to solve the time dependent heat equation in an arbitrary triangulated 2D region.

HOT_PIPE is a MATLAB program which uses FEM_50_HEAT to solve a heat problem in a pipe.

HOT_POINT is a MATLAB program which uses FEM_50_HEAT to solve a heat problem with a point source.

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.c, the source code.
• fd1d_heat_explicit.csh, commands to compile the source code.

Examples and Tests:

• x.txt a file containing the spatial coordinates.
• t.txt a file containing the times.
• u.txt a file containing the values of the computed solution at each point X and time T.
• solution.png a PNG image of the MATLAB graphics created by the commands:

              x = load ( 'x.txt' );
              t = load ( 't.txt' );
              u = load ( 'u.txt' );
              [ xg, tg ] = meshgrid ( x, t );
              mesh ( xg, tg, u );
            

List of Routines:

• MAIN is the main program for FD1D_HEAT_EXPLICIT.
• DTABLE_CLOSE_WRITE closes a file to which a DTABLE was to be written.
• DTABLE_DATA_WRITE writes data to a DTABLE file.
• DTABLE_HEADER_WRITE writes the header of a DTABLE file.
• DTABLE_WRITE writes information to a DTABLE file.
• F returns the right hand side of the heat equation.
• R8_EPSILON returns the R8 roundoff unit.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TIMESTRING returns the current YMDHMS date as a string.
• U0 returns the initial condition at the starting time.
• UA returns the Dirichlet boundary condition at the left endpoint.
• UB returns the Dirichlet boundary condition at the right endpoint.

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