Stability and Error Estimates for Convolution Quadrature under Radial Symbol Bounds

Abstract

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 $|F(z)|\leq CΦ(|z|)$ and consider two classes of radial functions: monotone bounds satisfying a scaling condition and regularly varying bounds with locally bounded above and away from zero. For strongly $A(α)$-stable multistep methods, we derive approximation and stability estimates for the CQ weights, as well as error bounds for power-type inputs $g_β(t)=t^{β-1}/Γ(β)$ 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 slowly varying and multiscale factors that would be lost under a coarser power majorant. The theory is applied to fractional viscoelastic models, distributed-order operators, and logarithmic-power memory operators. Numerical experiments with backward Euler and second-order backward differentiation CQ confirm the expected temporal convergence orders for direct responses. Complementary evolution examples demonstrate the corresponding temporal behavior for smooth and nonsmooth initial data.