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The analysis of the modified partial differential equation (MDE) of the constant wind speed advection equation explicit difference scheme up to the eighth order with respect to both space and time derivatives is presented. So far, in majority of publications this modified equation has been derived mainly as a fourth-order equation. The MDE is presented in the so-called  Π -form of the first differential approximation. This form includes only the space derivatives of higher order  p  and their coefficients  μ ( p ). Analysis of these coefficients provides indications of the nature of the dissipative and dispersive errors. A fragment of the stencil for determining the modified differential equation up to the eighth-order MDE for the second-order Beam–Warming scheme is included. The derived coefficients  μ ( p ) as well as the analysis of the phase shift errors, the phase and group velocities and dispersive features on the basis of these coefficients have not been published so far. The dissipative features of this method we present in [ 33 ].

High order modified differential equation of the Beam–Warming method, I. The dispersive features