An Alternative Numeric Scheme for Implicit Diffusion

To implement the standard implicit scheme for fixed boundary conditions, we need to solve a system of linear equations that looks like this (here written for only five samples in $x$, but the structure should be clear).

This is a specific case of the general tridiagonal system.

We can use Gaussian elimination to systematically remove all off-diagonal elements. First, to get rid of $a_2$, multiply the first row by $a_2/b_1$ and subtract it from the second.

We can write this as

where $b_2' = b_2 - c_1 a_2 / b_1$ and $y_2' = y2 - y_1 a_2 / b_1$.

To knock out $a_3$, we multiply the second row by $a_3 / b_2'$ and subtract from the third. Thus $b_3' = b_3 - c_2 a_3 / b_2'$, and $y_3' = y_3 - y_2' a_3 / b_2'$. After repeating this process for all remaining rows, we end up with

where

Now we remove the entries above the diagonal, starting with $c_4$. Multiply the last row by $c_4 / b_5'$ and subtract from the fourth. This doesn’t change $b_4'$, but it does yield $y_4'' = y_4' - y_5' c_4/ b_5'$. So after doing this $m - 1$ times (matrices shown for $m = 5$), we have

where

Now the solution is evident: $x_i = y_i'' / b_i'$.

For our particular system,

where

Plugging things in, we find that