With boundary conditions fixed at any unclosed curve (e.g., the satellite trajectory), both Poisson’s and Grad–Shafranov equations result in an ill-posed problem. This causes the exponentially growing perturbations analogous to those of Hadamar’s example. The suppression of these perturbations requires some regularization procedure. In our study we compared the efficiency of two regularization techniques, using either Savitzky-Golay filtering or the boundary layer approximation.
The benchmark reconstruction of the PIC simulations data has shown that the main contribution for inaccuracy arises from replacing Poisson’s equation by the equation of Grad–Shafranov. Under this substitution, the reachable cross-size of the reconstructed region is shrinking down to fractions of the proton inertial length. In terms of the reconstruction error, both regularization techniques perform nearly the same; the boundary layer approximation benefits from the comparative simplicity and less restrictions imposed on the boundary shape.