Difference stencils approximating Laplacian

Question

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.

Answer 1

You have assumed an equidistant grid in one dimension, but the answer below can be formulated (and is true) for general grids in any number of dimensions. You also haven't specified the boundary conditions, but as long as the boundary conditions lead to a well-posed problem, the result below applies (assuming a stable discretization of the boundary conditions).

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.

I'm not aware of a reference dealing directly with convergence of the semi-discrete system. But the result on convergence of the fully-discrete system implies the convergence of the semi-discrete system.

Note that the consistency condition can be written as an algebraic condition on the finite difference coefficients by using Taylor series.