In the present research paper we propose a new generator of families of numerical methods with increasing number of internal layers in an attempt to achieve higher order accuracy. The intermediate stages consist of predictor-corrector methods. The final layer is a symmetric two step method with constant coefficients and also free parameters. Those parameters define each family of methods. At first the method is constructed with unknown parameters and subsequently their value is estimated in order to fulfill the requirement of maximum phase-lag order. The stability of the new numerical algorithm is analyzed and the local truncation error is computed. The generator of the new families is applied to well known problems and is found to be more efficient compared to other methods and numerical methods generators with similar characteristics, which attempt to numerically solve such problems.

中文翻译：

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基于相位滞后泰勒级数的数值方法族发展算法

在本研究论文中，我们提出了一种新的数值方法族生成器，其内部层数不断增加，以试图实现更高的阶精度。中间阶段包括预测器-校正器方法。最后一层是具有常数系数和自由参数的对称两步法。这些参数定义了每个方法系列。该方法首先用未知参数构建，然后估计它们的值以满足最大相位滞后阶数的要求。分析了新数值算法的稳定性，计算了局部截断误差。新族的生成器应用于众所周知的问题，并且与具有相似特性的其他方法和数值方法生成器相比，效率更高，