A question about the accuracy of the gradient

[friedenhe]

Discussed in #714

^{Originally posted by WZDDZFJ December 4, 2024}
In the verification section, you introduced forward-mode differentiation to check the correctness of the gradients. Could you provide an example of using finite differences to directly check the correctness of the gradients? Because I followed the tutorial to analyze the two-dimensional airwing of NACA0012, but when I looked at the gradient file totalderivist.txt, I found that the FFD points in a symmetric position had different gradient values, and the gap was relatively small when it was not deformed, but the gap was large after deformation. For example, what I've framed in red is a typical set of symmetric FFD points. Usually, we use the finite difference method to verify the gradient, that is, manually give a disturbance to the FFD point, see how the objective function changes, and then obtain the gradient value. But in this case of the airfoil, do I perturb one FFD point at a time, or do I perturb a set of symmetric FFD points at the same time and then compare it to the gradient obtained by the discrete adjoint solution?

[friedenhe]

It seems that the derivs are not symmetric whenever you have non-zero FFD displacements.