This post is closely related to this one Centered Divided Difference approximations whose purpose is to build Finite-Difference (FD) approxmation schemes.
In this latter post I also talked, without explicitely naming it, about Truncation Error, see for instance https://en.wikiversity.org/wiki/Numerical_Analysis/Truncation_Errors.
I am foccusing here on a less known concept named "Equivalent Equation" (sometimes named "Modified Equation").
The seminal paper (no free acccess, which is surprising since it was first published 50 years ago) is this one by Warming and Hyett https://www.sciencedirect.com/science/article/pii/0021999174900114.
For a scholar example you can see here https://guillod.org/teaching/m2-b004/TD2-solution.pdf.
An alternative method to that of Warming and Hyett to derive the Equivalent Equation is given here (in French)
http://www.numdam.org/item/M2AN_1997__31_4_459_0.pdf (Carpentier et al.)
I never heard of the concept of Modified Equation applied to elliptic PDEs ; it's main domain of application is advection PDEs (parabolic PDEs in simpler cases).
Basically, any numerical scheme for solving an ODE or a PDE has a Truncation Error: this means there is some discrepancy between the exact solution of this PDE and the solution the scheme provides.
This discrepancy depends on the truncation error, in space alone for ODEs or elliptic PDEs, or in space and time for hyperbolic or advection PDEs.
One main problem with the Truncation Error is that it doesn't enable to understand the impact it will have on the solution it returns. For instance, will this sheme introduce diffusion, antidiffusion, scattering, ...
The aim of the Modified Equation is to answer these questions by constructing the continuous equation a given numerical scheme solves with a zero truncation error.
This is the original point of view Warming and Hyett developped in their 1974 paper.
This is a subject I work on 30 years ago (Maple V), and later in 2010.
It is very easy with Maple to do the painstaking development that Warming and Hyett did by hand half a century ago. And it is even so easy that the trick used by Carpentier et al. to make the calculations easier has lost some of its interest.
Two examples are given in the attched file
If a moderator thinks that this post should be merged with Centered Divided Difference approximations, he can do it freely, I won't be offended.