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


FD1D_HEAT_EXPLICIT, 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)
      
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 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
      
or, assuming we have solved for all values of U at time T, we have
       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 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_explicit_test

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:


Last revised on 01 March 2020.