The time fractional ODEs are equivalent to convolutional Volterra integral equations with completely monotone kernels. We introduce the concept of complete monotonicity-preserving ($\mathcal{CM}$-preserving) numerical methods for fractional ODEs, in which the discrete convolutional kernels inherit the $\mathcal{CM}$ property as the continuous equations. We prove that $\mathcal{CM}$-preserving schemes are at least $A(\pi / 2)$ stable and can preserve the monotonicity of solutions to scalar nonlinear autonomous fractional ODEs, both of which are novel. Significantly, by improving a result of Li and Liu (Quart. Appl. Math., 76(1):189-198, 2018), we show that the $\mathcal{L}1$ scheme is $\mathcal{CM}$-preserving. The good signs of the coefficients for such class of schemes ensure the discrete fractional comparison principles, and allow us to establish the convergence in a unified framework when applied to time fractional sub-diffusion equations and fractional ODEs. The main tools in the analysis are a characterization of convolution inverses for completely monotone sequences and a characterization of completely monotone sequences using Pick functions due to Liu and Pego (Trans. Amer. Math. Soc. 368(12): 8499-8518, 2016). The results for fractional ODEs are extended to $\mathcal{CM}$-preserving numerical methods for Volterra integral equations with general completely monotone kernels. Numerical examples are presented to illustrate the main theoretical results.

中文翻译：

---

时间分数 ODE 的完全保持单调性的数值方法

时间分数 ODE 等效于具有完全单调核的卷积 Volterra 积分方程。我们引入了分数 ODE 的完全单调性保持（$\mathcal{CM}$-preserving）数值方法的概念，其中离散卷积核继承了 $\mathcal{CM}$ 属性作为连续方程。我们证明 $\mathcal{CM}$-preserving 方案至少是 $A(\pi / 2)$ 稳定的，并且可以保持标量非线性自治分数 ODE 解的单调性，这两者都是新颖的。值得注意的是，通过提高 Li 和 Liu ( Quart. Appl. Math., 76(1):189-198, 2018)，我们证明 $\mathcal{L}1$ 方案是 $\mathcal{CM}$-preserving。此类方案的系数的良好迹象确保了离散分数比较原则，并允许我们在应用于时间分数子扩散方程和分数 ODE 时在统一框架中建立收敛。分析中的主要工具是完全单调序列的卷积逆表征和使用由于 Liu 和 Pego 的 Pick 函数的完全单调序列的表征（Trans. Amer. Math. Soc.368(12): 8499-8518, 2016)。分数 ODE 的结果扩展到 $\mathcal{CM}$-preserving 数值方法，用于具有一般完全单调核的 Volterra 积分方程。给出了数值例子来说明主要的理论结果。