To the best of my knowledge, the first rigorous approach was presented here
Article The Modified Equation Approach to the Stability and Accuracy...
and all studies followed that approach.
But I wonder what about the approach to follow in case of an integral equation that is discretized?
Have you ever seen some study wherein the modified integral equation is illustrated?
The same approach can be used, since an integral equation is solved numerically be transforming it into a system of equations, just as is done for a differential equation. I would have thought, however, that the stability criteria for finite-difference schemes were known before 1974...
Stam Nicolis
I am focusing only on the MDE topic, the stability analisys is due to von Neumann, many years before.
However, I consider that an integral equation is in weak form, there is no need to require regularity in space. Thus, we can introduce a Taylor series in time (provided the function is regular in time) but not in space. We can only integrate in space the Taylor series.
But I am not aware of some published paper about that.
Isn't it done here:
Article A Truncation Error Analysis of Third-Order MUSCL Scheme for ...
and probably also some of the papers that followed this one? I recalled of seeing a full 3D unstructured version in one of them, but I can't find it now
Paolo Lampitella
Paolo I know that paper and the answer is yes and not... The local truncation error is expressed in terms of pointwise derivatives but I mean an expression of the LTE wherein the integrals appear explicitly, congruently to the initial integral equation.
Ok, let me say that we could think to start from any MDE derived in a classic way for a PDE and integrate its LHS and RHS to generate the integral modified equation.
But if we start from the original idea that the weak equation does not require the differentiability, we should not introduce any Taylor expansion in space. Otherwise we should regularize the variable and define the Taylor expansion for the smooth averaged function.
It is possible to have LTE wherein the integrals appear explicitly, congruently to the initial integral equation. You can contact the book of Prof. Luigi Brugnano on line integral equations.