This is a modified Leapfrog method for the heat equation:
$u_{t} = u_{xx}$ in $\Omega = [0, T]$x$[0, 1]$
$u(0, x) = u_{0}(x)$
$u(t, 0) = g_{1}(t)$
$u(t, 1) = g_{2}(t)$
The "n" indices are time indices and the "i" indices are spatial indices.
$$\frac{u_i^{n+1} - u_i^{n-1}}{2 \, \Delta t} = \frac{u_{i-1}^n + u_{i+1}^n - u_i^{n-1} - u_i^{n+1}}{\Delta x^2}$$
Is this method consistent, stable and convergent?
Note that
$\frac{\frac{u_{i}^{n+1} - u_{i}^{n-1}}{2} }{\Delta t} = \frac{\frac{(u_{i}^{n+1} +u_i^{n}) - (u_{i}^{n-1}+u_i^{n})}{2}}{\Delta t} = \frac{u_i^{n+\frac{1}{2}}-u_i^{n-\frac{1}{2}}}{\Delta t}$ which is the centered finite difference for the first t partial derivative $O(\Delta t ^2)$ (1).
And $u^n_{i-1} - u^{n-1}_i = 2\frac{u^n_{i-1} +u^n_{i}-(u^{n-1}_i + u^{n}_i)}{2} = 2u^n_{i}$, then $$\frac{u^n_{i-1}+u^n_{i+1} - u^{n-1}_{i} - u^{n+1}_{i}}{\Delta x^2} = \frac{u^n_{i-1}+2u^{n}_{i} - u^{n+1}_{i}}{\Delta x^2} $$ is the centered finite difference for the x second derivative which is $O(\Delta x^2)$. (2)
Now we can conclude that the truncation error order is $O(\Delta t^2, \Delta x^2)$ and the method is consistent. Using (1) and (2) you may expand it in taylor series and conclude the same. Using von Neumman analysis we may show that the method is unconditionally stable. Since it is consistent and stable, it is convergent.