# fd1d_heat_implicit

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.

### Languages:

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.

### Related Data and codes:

FD1D_BURGERS_LAX, a FORTRAN90 code 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 FORTRAN90 code 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 FORTRAN90 code which applies the finite difference method to a two point boundary value problem in one spatial dimension.

FD1D_HEAT_EXPLICIT, a FORTRAN90 code 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 FORTRAN90 code which uses the finite difference method to solve the steady (time independent) heat equation in 1D.

FD1D_PREDATOR_PREY, a FORTRAN90 code which uses finite differences to solve a 1D predator prey problem.

FD1D_WAVE, a FORTRAN90 code which applies the finite difference method to solve the time-dependent wave equation utt = c * uxx in one spatial dimension.

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

FEM1D, a FORTRAN90 code which applies the finite element method, with piecewise linear basis functions, to a linear two point boundary value problem;

FEM2D_HEAT, a FORTRAN90 code which applies the finite element method to solve the 2D heat equation.

### 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 29 June 2020.