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].

中文翻译：

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Beam-Warming方法的高阶修正微分方程，I.色散特征

摘要 对恒风速平流方程显式差分格式的修正偏微分方程(MDE) 进行了分析，分析了空间导数和时间导数的八阶。到目前为止，在大多数出版物中，这个修改后的方程主要是作为四阶方程推导出来的。MDE 以所谓的一阶微分逼近的 Π 形式表示。这种形式仅包括高阶 p 的空间导数及其系数 μ(p)。这些系数的分析提供了对耗散和色散误差性质的指示。包括用于确定二阶光束加热方案的八阶 MDE 的修正微分方程的模板片段。迄今为止，导出的系数 μ(p) 以及基于这些系数的相移误差、相速度和群速度以及色散特征的分析尚未公布。我们在 [33] 中介绍了这种方法的耗散特性。