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The paper presents an algorithm to simulate the electric corona discharge in the two-dimensional hyperbolic needle-ground plate configuration. The boundary conditions for the space charge density are derived from the direct ionisation criterion. This technique doesn't rely on the Kaptzov hypothesis and can be used for any shape of the discharge electrode. The proposed algorithm has been compared with two approximate ones. One uses the Kaptzov hypothesis that the electric field on the corona electrode surface remains constant, with the value derived from the ionisation criterion, assuming that the electric field is Laplacian. The other one mechanically applies the Peek formula for the cylindrical wire, after replacing the wire radius with the surface curvature radius at given point of the electrode. The differences between all three approaches are negligible for small corona currents; they are noticeable, but still small, for higher currents.