So, I am deriving a high-order compact finite difference scheme, and got into the equation below:
\begin{multline} \delta t^2 \{1 + \dfrac{h^2}{12} (\delta x^2 + \delta y^2 + \delta z^2) + \dfrac{h^4}{360} [(\delta x^4 + \delta y^4 + \delta z^4) - (\delta x^2 \delta y^2 + \delta x^2 \delta z^2 + \delta y^2 \delta z^2)]\} \Psi_{i,j,k}^n = \\ F(\delta x^2, \delta y^2, \delta z^2) \Psi_{i,j,k}^n + \mathcal{O}(h^6), \end{multline}
where $\delta x^2$, $\delta y^2$ and $\delta z^2$ are central finite difference operators for the second derivative, $F$ is a function dependent on this operators, $h$ is grid distance to the discrete version of the problem, and $\Psi(x,y,z,t) \rvert_{i,j,k}^n = \Psi_{i,j,k}^n$ is a function dependent on position and time. Can I truncate only the left side of this equation to the first term, i.e., to $\delta t^2 \Psi_{i,j,k}^n$, and keep the right side as it is?
My derivation is just like the ones made in papers "High-Order Compact Finite Difference Methods" and "A High-Order Compact Formulation for the 3D Poisson Equation" by Spotz & Carey.
If you truncate the LHS as described, you will end up with an $O(h^2)$ scheme. If you are OK with that, that's fine, but why are you exploring an HOC scheme? The $h^2/12$ term on the LHS has readily available finite difference approximations, and if you include that in your overall approximation, you immediately bump up the accuracy to $O(h^4)$. The downside, perhaps, is that you force yourself into an implicit method, but the upside is that it should be guaranteed to be stable. The references you cite might also attempt to approximate the $h^4/360$ terms by differentiating the governing equation, but only if 6th-order accuracy is desired.