# Difference stencils approximating Laplacian

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 believe this question is too basic to be appropriate for MO, but it is very well written and I will give a short answer in any case. – David Ketcheson Nov 18 '15 at 7:51

The necessary and sufficient condition is that the semi-discretization $\Delta^h$ be consistent. In other words, given a twice-differentiable function $u(x)$, let $u^h$ denote the vector of values of $u(x)$ evaluated at the points $x_i$. Then $\Delta^h$ is said to be consistent if $$\lim_{h\to 0} \left((\Delta^h u^h)_i - \Delta u(x_i)\right) = 0.$$ Any consistent $\Delta^h$ will give an ODE system whose solution converges to that of the heat equation.
In fact, we can make an even stronger statement. If $\Delta^h$ is consistent and if the resulting ODE system is discretized in time in a consistent, stable way, then the fully-discrete solution will converge to that of the heat equation provided the CFL condition is satisfied. For details, see the original paper of Courant, Friedrichs, and Lewy (which doesn't actually cover parabolic equations, but the tools are all there) and the paper of Lax and Richtmyer. For a more modern treatment, I recommend chapter 9 of LeVeque's book.