In this paper, we investigate and analyze numerical solutions for the Volterra integrodifferential equations with tempered multi-term kernels. Firstly we derive some regularity estimates of the exact solution. Then a temporal-discrete scheme is established by employing Crank-Nicolson technique and product integration (PI) rule for discretizations of the time derivative and tempered-type fractional integral terms, respectively, from which, nonuniform meshes are applied to overcome the singular behavior of the exact solution at $t=0$. Based on deduced regularity conditions, we prove that the proposed scheme is unconditionally stable, and possesses accurately temporal second-order convergence in $L_2$-norm. Numerical examples confirm the effectiveness of the proposed method.

翻译：在本文中，我们研究并分析了采用渐进多项式核的Volterra积分微分方程的数值解。首先我们推导了准确解的一些正则性估计。然后利用Crank-Nicolson方法和乘积积分(PN)规则，分别对时间导数和温和型分数积分术进行离散化，建立了时间离散方案，其中，采用非均匀网格来克服在 $t=0$ 处的精确解的奇异行为。基于推导出的正则条件，我们证明了所提出的方案是无条件稳定的，并且在 $L_2$ -范数下具有准确的时间二阶收敛性。数值实验验证了所提出方法的有效性。