Implicit block approaches are used by a number of numerical analyzers to model mild, medium, and hard differential systems. Their excellent stability characteristics, self-starting nature, quick convergence, and large decrease in computing cost all contribute to their widespread application. With these numerical benefits in mind, a new one-step implicit block method with three intrastep grid points has been created. The major term of the local truncation error is minimized to determine which of these points is optimal. The reformulation of the suggested technique leads to a significant decrease in computing cost while maintaining the same consistency, zero-stability, $\mathcal{A}$ -stability, and convergence. Several sorts of error are calculated, together with CPU time and efficiency plot, to determine which is superior. Differential models from the fields of heat transfer, population dynamics, and chemical engineering show that the suggested method does a better job than some of the current hybrid block and implicit Radau methods with similar properties.

中文翻译：

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一种通过插值和搭配获得最优步内点的连续混合分块方法

许多数值分析器使用隐式块方法来模拟轻度、中度和硬微分系统。它们优异的稳定性特性、自启动性、收敛速度快、计算成本大幅降低等特点，都有助于它们的广泛应用。考虑到这些数值优势，创建了一种新的具有三个步内网格点的单步隐式块方法。局部截断误差的主要项被最小化以确定这些点中的哪一个是最优的。建议技术的重新制定导致计算成本显着降低，同时保持相同的一致性、零稳定性、$\mathcal{A}$ -稳定性和收敛性。计算了几种错误，连同 CPU 时间和效率图，以确定哪个更好。