Abstract Production-destruction systems (PDS) of ordinary differential equations (ODEs) are used to describe physical and biological reactions in nature. The considered quantities are subject to natural laws. Therefore, they preserve positivity and conservation of mass at the analytical level. In order to maintain these properties at the discrete level, the so-called modified Patankar-Runge-Kutta (MPRK) schemes are often used in this context. However, up to our knowledge, the family of MPRK has been only developed up to third order of accuracy. In this work, we propose a method to solve PDS problems, but using the Deferred Correction (DeC) process as a time integration method. Applying the modified Patankar approach to the DeC scheme results in provable conservative and positivity preserving methods. Furthermore, we demonstrate that these modified Patankar DeC schemes can be constructed up to arbitrarily high order. Finally, we validate our theoretical analysis through numerical simulations.

中文翻译：

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任意高阶、保守和正性保留的帕坦卡型延迟校正方案

摘要 常微分方程 (ODE) 的生产-破坏系统 (PDS) 用于描述自然界中的物理和生物反应。所考虑的数量受自然法则的约束。因此，它们在分析水平上保持了质量的正性和守恒。为了在离散级别保持这些属性，在这种情况下经常使用所谓的改进的 Patankar-Runge-Kutta (MPRK) 方案。然而，据我们所知，MPRK 家族仅发展到三阶精度。在这项工作中，我们提出了一种解决 PDS 问题的方法，但使用延迟校正 (DeC) 过程作为时间积分方法。将修改后的 Patankar 方法应用于 DeC 方案会产生可证明的保守和积极性保持方法。此外，我们证明了这些修改后的 Patankar DeC 方案可以构建到任意高阶。最后，我们通过数值模拟验证了我们的理论分析。