The Deferred Correction (DeC) is an iterative procedure, characterized by increasing accuracy at each iteration, which can be used to design numerical methods for systems of ODEs. The main advantage of such framework is the automatic way of getting arbitrarily high order methods, which can be put in Runge--Kutta (RK) form. The drawback is the larger computational cost with respect to the most used RK methods. To reduce such cost, in an explicit setting, we propose an efficient modification: we introduce interpolation processes between the DeC iterations, decreasing the computational cost associated to the low order ones. We provide the Butcher tableaux of the new modified methods and we study their stability, showing that in some cases the computational advantage does not affect the stability. The flexibility of the novel modification allows nontrivial applications to PDEs and construction of adaptive methods. The good performances of the introduced methods are broadly tested on several benchmarks both in ODE and PDE contexts.

翻译：延迟校正（DeC）是一种迭代过程，其特点是在每次迭代中精度逐步提高，可用于设计常微分方程（ODE）系统的数值方法。该框架的主要优势在于能自动生成任意高阶方法，且可转化为龙格-库塔（Runge-Kutta, RK）形式。其缺点是与常用RK方法相比计算成本较大。为降低此类成本，在显式设定下我们提出一种高效改进：在DeC迭代之间引入插值过程，从而减少与低阶迭代相关的计算量。我们给出了新修正方法的Butcher表，并研究了其稳定性，表明在某些情况下计算优势不会影响稳定性。该新型修正的灵活性使其能够应用于偏微分方程（PDE）及自适应方法的构建。通过常微分方程与偏微分方程领域的多个基准算例，对所提方法的优异性能进行了广泛测试。