Order Stars, Approximations and Finite Differences. III Finite Differences for $u_t = \omega u_{xx} $

Given the finite difference discretization \[\sum_{j = - r}^r {\alpha _j (\mu )U_{m + j}^{n + 1} = \sum_{j = - r}^r {\beta _j (\mu )U_{m + j}^n } ,\quad \mu = \frac{{\omega \Delta t}}{{(\Delta x)^2 }}} .\] of the differential equation $u_t = \omega u_{xx} ,\omega \in \mathbb{C}$, $\operatorname{Re} \omega \geqq 0$, we prove that a unique choice of coefficients gives order $4r + 1$ and that no higher order method exists. Furthermore, we show that this highest order method is stable for every $r \geqq 1$. Our analysis uses order stars of first and second kind, in conjunction with Padé theory and computational complex analysis.