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

FD1D_HEAT_IMPLICIT is a MATLAB 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)

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)

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

or, assuming we have solved for all values of U at time T, we have

            -     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)

which can be written as A*x=b, where A is a tridiagonal matrix whose entries are the same for every time step.

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 FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

FD1D_BURGERS_LAX, a MATLAB 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 MATLAB 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 MATLAB program which applies the finite difference method to a two point boundary value problem in one spatial dimension.

FD1D_HEAT_EXPLICIT, a MATLAB program which uses the finite difference method to solve the time dependent heat equation in 1D, using an explicit time step method.

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

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

FD1D_WAVE, a MATLAB program which applies the finite difference method to solve the time-dependent wave equation in one spatial dimension.

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

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

FEM2D_HEAT, a MATLAB program which applies the finite element method to solve the 2D heat equation.

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

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

Reference:

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_cfl.m, computes the Courant-Friedrichs-Loewy coefficient;
fd1d_heat_implicit.m, carries out the calculation;
r8mat_write.m, writes an R8MAT to a file.
r8vec_write.m, writes an R8VEC to a file.
timestamp.m, prints the YMDHMS date as a timestamp;

Examples and Tests:

fd1d_heat_implicit_test.m, runs all the tests.
fd1d_heat_implicit_test01.m, runs test 1;
fd1d_heat_implicit_test02.m, runs test 2;
fd1d_heat_implicit_test03.m, runs test 3;
fd1d_heat_implicit_test_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.

bc_test01.m, enforces the boundary conditions.
h_test01.txt, the computed H data.
ic_test01.m, enforces the initial condition.
k_test01.m, sets the conductivity.
plot_test01.png, a PNG image of the solution.
rhs_test01.m, supplies the right hand side source term.
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 ).

bc_test02.m, enforces the boundary conditions.
exact_test02.m, evaluates the exact solution.
g_test02.txt, the exact data.
h_test02.txt, the computed H data.
ic_test02.m, enforces the initial condition.
k_test02.m, sets the conductivity.
plot_test02.png, a PNG image of the solution.
rhs_test02.m, supplies the right hand side source term.
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.

bc_test03.m, enforces the boundary conditions.
h_test03.txt, the computed H data.
ic_test03.m, enforces the initial condition.
k_test03.m, sets the conductivity.
plot_test031.png, a PNG image of the solution.
rhs_test03.m, supplies the right hand side source term.
t_test03.txt, the T data.
x_test03.txt, the X data.

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

Last revised on 31 January 2012.