In this work, we further investigate the application of the well-known Richardson extrapolation (RE) technique to accelerate the convergence of sequences resulting from linear multistep methods (LMMs) for numerically solving initial-value problems of systems of ordinary differential equations. By extending the ideas of our previous paper, we now utilize some advanced versions of RE in the form of repeated RE (RRE). Assume that the underlying LMM -- the base method -- has order $p$ and RE is applied $l$ times. Then we prove that the accelerated sequence has convergence order $p+l$. The version we present here is global RE (GRE, also known as passive RE), since the terms of the linear combinations are calculated independently. Thus, the resulting higher-order LMM-RGRE methods can be implemented in a parallel fashion and existing LMM codes can directly be used without any modification. We also investigate how the linear stability properties of the base method (e.g. $A$- or $A(\alpha)$-stability) are preserved by the LMM-RGRE methods.

翻译：本文进一步研究了著名的Richardson外推（RE）技术在加速线性多步法（LMM）序列收敛性方面的应用，该序列产生于数值求解常微分方程组初值问题。通过扩展我们先前论文的思想，我们现在采用重复RE（RRE）形式的高级RE版本。假设基础LMM——即基方法——具有$p$阶精度，且RE被应用$l$次。我们证明加速后的序列具有$p+l$阶收敛精度。这里提出的版本是全局RE（GRE，亦称为被动RE），因为线性组合的各项是独立计算的。因此，所得到的高阶LMM-RGRE方法可以并行实现，且现有的LMM代码无需任何修改即可直接使用。我们还研究了基方法的线性稳定性性质（例如$A$-稳定性或$A(\alpha)$-稳定性）如何被LMM-RGRE方法所保持。