So I had a chance to write down the finite difference equations to convert the partial operators into linear algebra form.
\begin{aligned}
u_t + k u_{xx} &= h (x,t) \\
u_t &= \frac{u_i^{n+1}-u_i^n}{\Delta t}\\
u_{xx} &= \frac{u_{i+1}^{n+1}-2u_i^{n+1}+u_{i-1}^{n+1}}{\Delta x^2}\\
\frac{u_i^{n+1}-u_i^n}{\Delta t} + k\frac{u_{i+1}^{n+1}-2u_i^{n+1}+u_{i-1}^{n+1}}{\Delta x^2} &= h_i^n\\
\end{aligned}
This is obviously with an implicit euler integration. The thing to note is the stencil, where the n index of h is related to n and n+1 (possibly
even n-1 in general depending on any other schemes to approximate the first partial derivative), and the same for each i index related to i, i-1, and
i+1. If u and h are both spatio-temporal where column n is the instantaneous solution at time n, then they are both matrices. The L operator
operates on a matrix and produces a matrix, but in this way it isn't standard linear algebra since that operates vector by vector and this stencil has
to operate between columns. No L would take the 4th order form:
\begin{aligned}
L_{jkmn}u_{mn} &= h_{jk}
\end{aligned}
Or multilinear algebra. Sadly after Einstein, it has only had sporadic work done because of the computational cost until most recently with big data
mining. I'm currently reading some recent articles to find what modern mathematicians are writing about multilinear spaces. |
