Suppose I want to use the 4-th order accurate central finite-difference scheme for a boundary value problem with the second derivative on regularly spaced nodes in $[0, 1]$, with (homogeneous) Dirichlet boundary conditions. Say I have 5 interior nodes, $\mathbf{u}$. My differential operator matrix becomes:
$$ \mathbf{L} \mathbf{u} = \begin{bmatrix} c & b & a & & \\ b & c & b & a & \\ a & b & c & b & a \\ & a & b & c & b \\ & & a & b & c \\ \end{bmatrix} \mathbf{u}, $$
which arises by setting
$$u(0) = u(1) = 0 \qquad (2)$$
and
$$u(0-dx) = u(1+dx) = 0. \qquad (3)$$
Note that the latter (ghost point) conditions are necessary because of the higher-order scheme. However, I'm not sure if this use of ghost points is quite appropriate because, in effect, I'm fixing $u$ over intervals, rather than at points. Indeed, if I compute the eigenvalue decomposition of this matrix, the eigenvectors differ slightly, but significantly, from their theoretical counterparts (which they do not if use the simpler 2nd order scheme).
Of course, I could instead reduce the accuracy of the finite-difference scheme near the boundaries, Or I could use a non-central scheme near the boundaries. However, both of those solutions would ruin the symmetry of $\mathbf{L}$, which I find undesirable (for example for efficient inversion).
So what is the best way to go about this? Is my use of ghost points really appropriate?
Edit1:
I believe that instead of using (3) to specify the ghost points, I should impose the Dirichlet conditions on the average around the boundaries: $$ \begin{align*} \frac{u(-dx) + u(+dx)}{2} = 0, \qquad (4) \\ \text{i.e.} \quad u_{-1} = - u_{1} \qquad (5) \end{align*} $$ And in the case of even higher-order central differences, $u_{-n} = - u_{n}$, for $n$ between 1 and the number of ghost points. I've checked, and this produces an operator matrix that is symmetric, and whose eigenvectors equal the theoretical ones. So I guess my question becomes: does anybody have a name and theoretical reference for this approach?
