The mathematical modeling of various real life applications leads to systems of ordinary differential equations which include crucial properties like the positivity of the solution as well as the conservation of mass or energy. Based on the fundamental work of Burchard et al. (2003), unconditionally positive and conservative modified Patankar–Runge–Kutta schemes (MPRK) are available. These methods are highly stable and often outperform standard Runge–Kutta schemes.

In this article, we extend MPRK methods, named MPRKO methods, using Oliver’s approach (Oliver, 1975) to improve the accuracy of these schemes in the field of nonautonomous systems. The approach does not demand $\mathbf{A}\mathbf{e}=\mathbf{c}$ in the Butcher tableau $(\mathbf{A},\mathbf{b},\mathbf{c})$, where $\mathbf{e}={(1,\dots ,1)}^T$. Following the general analysis of MPRK schemes described in Kopecz and Meister (2018), positivity and mass conservation fundamental properties are proven and even conditions concerning the Patankar weights are given to get second order accuracy of the MPRKO methods. Finally, we consider different linear models and a non-linear epidemiological SEIR problem to confirm the theoretical results and to give reliable statements about the accuracy of the novel class of MPRKO methods.

各种现实生活应用程序的数学建模导致了常微分方程系统的出现，这些系统包括一些关键特性，例如溶液的正性以及质量或能量的守恒。基于Burchard等人的基础工作。（2003年），无条件积极和保守的修正帕坦卡-龙格-库塔计划（MPRK）可用。这些方法非常稳定，并且通常优于标准的Runge-Kutta方案。

在本文中，我们使用奥利弗的方法（Oliver，1975）扩展了名为MPRKO方法的MPRK方法，以提高这些方案在非自治系统领域的准确性。该方法不要求$\mathbf{一种}\mathbf{Ë}=\mathbf{C}$ 在屠夫画面 $（\mathbf{一种}，\mathbf{b}，\mathbf{C}）$，在哪里 $\mathbf{Ë}={（\mathrm{1个}，\dots ，\mathrm{1个}）}^{Ť}$。根据Kopecz和Meister（2018）中描述的MPRK方案的一般分析，证明了阳性和质量守恒的基本性质，甚至给出了有关Patankar权重的条件，以获得MPRKO方法的二阶准确性。最后，我们考虑不同的线性模型和非线性流行病学SEIR问题，以确认理论结果并就新颖的MPRKO方法的准确性给出可靠的陈述。