Stability and Error Estimates for Convolution Quadrature under Radial Symbol Bounds
We develop stability and error estimates for multistep convolution quadrature (CQ) under general radial bounds on the Laplace symbol. Instead of the classical single-power condition $|F(z)|\leq M|z|^{-μ}$, we assume $|F(z)|\leq CΦ(|z|)$, where $Φ$ is positive, continuous and regularly varying at zero and infinity. For strongly $A(α)$-stable multistep methods, we derive error estimates and stability estimates for the CQ weights, as well as error bounds for power-type inputs $g_β(t)=t^{\beta1}/Γ(β)$ with $β>0$. These estimates explicitly retain the dependence on the memory law through the factor $Φ(t_n^{-1})$. The framework accommodates symbols whose behavior cannot be described by a global single-power. It also preserves logarithmic and multiscale factors that would be lost under a coarser power bounds. Then we apply the to fractional viscoelastic, distributed-order, and logarithmic-power memory models. Numerical experiments with backward Euler and second-order backward differentiation CQ confirm the expected temporal convergence orders for direct responses. A distributed-order PDE further illustrates temporal convergence for smooth and nonsmooth initial data.