Let $\Delta$ be the Laplace operator on the interval $[0,1]\subset \mathbb{R}$. Divide $[0,1]$ into small intervals of size $h$ to get an equidistant grid. One can approminate $-\Delta$ on this grid by $$ \frac{u_{i-1} - 2u_i + u_{i-1}}{h^2} $$ or $$ \frac{- u_{i+2} + 16 u_{i+1} - 30u_i + 16 u_{i-1} - u_{i-2}}{12h^2} $$ where the weights -30, 16, -1 are calculated from Taylor approximations with radius $h$ and $2h$ around a point, and optimized in order to get a higher accuracy, in the 1d case $O(h^4)$.

But we could also take wider stencils, and they might not be symmetric.

**My question**:Which are the (minimal) conditions on a finite difference stencil $\Delta^h$ such that in the limit as $h \to 0$, for sufficiently regular initial condition, a solution $u^h$ to the ODE system (discrete in space, continuous in time heat equation)
$$
\partial_t u^h(t) = \Delta^h u^h(t) \qquad t \in [0,T]
$$
converges to a solution $u$ to the heat equation on $[0,1] \times [0,T]$

**My thoughts:**

- $\Delta^h$ must be diagonally dominant and negative semidefinite
- $\Delta^h$ must satisfy a discrete coercivity condition or a strong monotonicity condition

I am grateful for any hints/comments/literature advice.