New approach to the Lax‐Wendroff modified differential equation for linear and nonlinear advection

Abstract

The paper presents an enhanced analysis of the Lax-Wendroff difference scheme-up to the eighth-order with respect to time and space derivatives-of the modified-partial differential equation (MDE) of the constant-wind-speed advection equation. The modified equation has been so far derived mainly as a fourth-order equation. The Π-form of the first differential approximation (differential approximation or equivalent equation) derived by expressing the time derivatives in terms of the space derivatives is used for presenting the MDE. The obtained coefficients at higher order derivatives are analyzed for indications of the character of the dissipative and dispersive errors. The authors included a part of the stencil applied for determining the modified differential equation up to the eighth-order of the analyzed modified differential equation for the second-order Lax-Wendroff scheme. Neither the derived coefficients at the space derivatives of order p ∈ (7 -8) in the modified differential equation for the Lax-Wendroff difference scheme nor the results of analyses on the basis of these coefficients of the group velocity, phase shift errors, or dispersive and dissipative features of the scheme have been published. The MDEs for 2 two-step variants of the Lax-Wendroff type difference schemes and the MacCormack This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Wstęp

“…Finally, analyzing Figure 14 we come to the conclusion that the spectral features of the Knighting (1955) and Ogura (1958) mask ( 12) is the worst (graph (a)). In numerical methods, a difference scheme characterized by backward diffusion is said to be an antidissipative one and, in general, it amplifies non-physical solutions (see: Winnicki et al (2019)). Figure 3b, Figure 12 (red dot-dashed line) and Figure 14a confirm this observation.…”

“…These mixed forms are called the Γ-forms. For isolating the terms responsible for numerical dispersion and dissipation it is necessary to express all time derivatives as spatial ones (see: Appadu et al, 2008;Appadu and Dauhoo, 2011;Appadu et al, 2014;Winnicki et al, 2019;Shokin et al, 2020).…”

“…A short history of the modified differential equations and the specialist terminology used by scientists from various countries can be found in Lerat and Peyret (1973), Warming and Hyett (1974), Peyret and Taylor (1983), Li and Yang (2011), Krawczyk et al (2012), Li and Yang (2013), Winnicki et al (2019), Shokin et al (2020).…”

Advances in Geodesy and Geoinformation

“…For the elliptic partial differential equations we always obtain their Π-forms. More work on the MDE can be found in Warming and Hyett (1974), Peyret and Taylor (1983), Li and Yang (2011), Winnicki et al (2019), Shokin et al (2020a, b).…”

Modified equation based mesh adaptation algorithm for evolutionary scalar partial differential equations

“…Durran [6], Mortan and Mayer [20] and Pletcher et al [21] limited the Lax-Wendroff modified differential equation up-to fourth-order for analyzing of difference scheme's features. Recently, Winnicki et al [32] derived the eighth order modified differential equation for the Lax-Wendroff difference scheme and analyze the dispersion, leading and lagging phase shift error and group velocity.…”