Projective Integration methods are explicit time integration schemes for stiff ODEs with large spectral gaps. In this paper, we show that all existing Projective Integration methods can be written as Runge-Kutta methods with an extended Butcher tableau including many stages. We prove consistency and order conditions of the Projective Integration methods using the Runge-Kutta framework. Spatially adaptive Projective Integration methods are included via partitioned Runge-Kutta methods. New time adaptive Projective Integration schemes are derived via embedded Runge-Kutta methods and step size variation while their accuracy, stability, convergence, and error estimators are investigated analytically and numerically.

翻译：投影积分方法是针对具有大谱间隙的刚性常微分方程的一种显式时间积分格式。本文证明所有现有投影积分方法均可表示为包含多个阶段的扩展布彻表形式的龙格-库塔方法。我们利用龙格-库塔框架证明了投影积分方法的一致性和阶条件。通过分区龙格-库塔方法引入了空间自适应投影积分方法。基于嵌入式龙格-库塔方法和步长变化推导出了新型时间自适应投影积分格式，并从解析和数值两个角度研究了其精度、稳定性、收敛性和误差估计量。