fd1d_heat_implicit, a FORTRAN90 code 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 code 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.
The computer code and data files described and made available on this web page are distributed under the MIT license
fd1d_heat_implicit is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.
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