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

Abstract

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].

CTCS Schemes for Second Order Wave Equation: Numerical Results and Spectral Analysis

“…Even-order derivatives lead to numerical dissipation or anti-dissipation depending on the sign of the coefficients. Odd-order derivatives in the modified equation indicates numerical dispersion [8,9,10].…”

“…The RPE is a ratio that measures the numerical wave velocity to that of the physical wave and gives us the necessary information to analyse dispersion caused by a numerical scheme [4]. Some work on modified equations of partial differential equations are detailed in [5,6,7,8,9]. In [5], an implicit finite-difference scheme is constructed for numerical solution of nonlinear hyperbolic systems in conservation law form.…”

“…Winnicki et al [7] obtained modified equations of the Lax-Wendroff scheme up to the eight order with respect to time and space derivatives for the constant-wind-speed advection equation. Shokin et al [8] analysed the modified PDE of the constant wind speed advection equation for an explicit difference scheme up to the eight order with respect to both space and time derivatives. The modified equation includes only the space derivatives of higher order p and their coefficients μ(p).…”