Year: 2026
Type: preprint
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 $|F(z)|\leq CΦ(|z|)$, 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.
Source: arXiv (Cornell University)
Authors: Ren, Jingchuang, Su, Haibin
Institution: Tsinghua University
Language: English
Subfield: Modeling and Simulation
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222982657