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

Optimal error estimate of accurate second-order scheme for Volterra integrodifferential equations with tempered multi-term kernels