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The way of approximating the gradient of the functional in a gradient descent method is question of interest. It is clear that the higher order of approximation is your numerical scheme, the better accuracy you get.. However, when solving backwards the adjoint equation it is not evident whether to choose the same order of approximation than the forward model or not in order to get the sensitivity of the system and consequently, the appropriate descent direction.
In order to leave aside the uncertainties related to more complex phenomena, non-linearities... the 2D linear scalar equation is used with two possible numerical schemes: First Order Upwind (FOU) and Second Order Upwind (SOU).
Two test cases with analytical solution are examined in order to determine the convenience of using one or another numerical scheme for the resolution of the adjoint equation. Video 1 shows the convergence when refining the mesh for the rotating cone test case for the FOU scheme. Video 2 and Video 3 display the comparison between FOU and SOU schemes respectively for the adjoint equation in the frontogenesis test case.
