Discrete L1 remainder stability of first and second order schemes for a Volterra integro-differential equation

This work investigates the discrete ??????1 remainder stability of first and second order schemes for a Volterra integro-differential equation, where the backward Euler and backward difference formula schemes are applied in combination with first and second order convolution quadrature for approximating the integral term, respectively. By Hardy's inequality and complex analysis techniques to accommodate the singularity of generating functions, we prove the long-time ??????1 remainder stability of numerical solutions of two schemes, which characterizes the long-time behavior of numerical solutions and indicates the preservation of asymptotically periodic stability of numerical schemes as continuous problems. Numerical experiments are performed to substantiate the theoretical analysis.