FEM1D_HEAT_EXPLICIT
Finite Element Solution of the
Time Dependent 1D Heat Equation
using Explicit Time Stepping

FEM1D_HEAT_EXPLICIT, a MATLAB library which solves the time-dependent 1D heat equation, using the finite element 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)
      

The spatial derivatives are approximated using the finite element method, with piecewise linear elements.

The solver applies an explicit forward Euler approximation to the first derivative in time.

Licensing:

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

Languages:

FEM1D_HEAT_EXPLICIT is available in a MATLAB version and a Python version.

Related Data and Programs:

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

fem1d_heat_explicit_test

FEM1D_HEAT_IMPLICIT, a MATLAB program which uses the finite element method and implicit time stepping with the backward Euler method to solve the time dependent heat equation in 1D.

FEM1D_HEAT_STEADY, a MATLAB program which uses the finite element method to solve the steady (time independent) heat equation in 1D.

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:

• assemble_fem.m, assembles the finite element stiffness matrix and right hand side.
• assemble_mass.m, assembles the finite element mass matrix.
• basis_function.m, evaluates the linear basis functions.
• fem1d_heat_explicit.m, takes one explicit time step.
• fem1d_heat_explicit_cfl.m, computes the Courant-Friedrichs-Loewy coefficient;
• quadrature_set.m, sets the quadrature rule;
• r8mat_write.m, writes an R8MAT to a file.
• r8vec_write.m, writes an R8VEC to a file.
• reference_to_physical.m, maps reference data to a subinterval;
• timestamp.m, prints the YMDHMS date as a timestamp;