Convolution Quadrature with Adaptive History Compression for Variable-Order Fractional Reaction--Diffusion Equations
Abstract
We develop a fully implicit fast method for semilinear variable-order fractional reaction--diffusion equations with periodic fractional diffusion and exponential-transition Scarpi memory. A Volterra-resolvent analysis establishes local well-posedness and temporal regularity estimates that describe the weak initial singularity. Backward Euler convolution quadrature combined with Fourier pseudospectral discretization achieves mesh-uniform stability and positive-time first-order convergence. To accelerate the scheme, we utilize Cauchy--FFT weight recovery and adaptive blockwise sum-of-exponentials compression, which significantly reduces the cost of history evaluation under local error control. A logarithmic perturbation bound quantifies the effects of weight recovery, history compression, and inexact nonlinear solves, giving tolerance conditions that preserve the convergence order. Numerical experiments confirm the first-order accuracy and demonstrate that the accelerated method efficiently resolves long-time spatial patterns at a substantially reduced computational cost.
Related
- Stochastic fractional diffusion equations with spatial Gaussian noise in the Stratonovich regime
- Data-driven accelerated multigrid method for Riesz space fractional diffusion equations
- Inverse source problem of variable-exponent time-fractional diffusion equations
- Stability and Error Estimates for Convolution Quadrature under Radial Symbol Bounds
- Highly Accurate Galerkin Approach for Modified Anomalous Time-Fractional Sub-Diffusion Equations Based on Fibonacci Coefficient Polynomials
BibTeX
@misc{yuan2026convolution,
title = {{Convolution Quadrature with Adaptive History Compression for Variable-Order Fractional Reaction--Diffusion Equations}},
author = {Pu Yuan and Paul Zegeling},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27262},
eprint = {2610.27262},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27262},
}
RIS
TY - UNPB
TI - Convolution Quadrature with Adaptive History Compression for Variable-Order Fractional Reaction--Diffusion Equations
AU - Pu Yuan
AU - Paul Zegeling
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.27262
UR - https://arxiv.org/abs/2610.27262
AB - We develop a fully implicit fast method for semilinear variable-order fractional reaction--diffusion equations with periodic fractional diffusion and exponential-transition Scarpi memory. A Volterra-resolvent analysis establishes local well-posedness and temporal regularity estimates that describe the weak initial singularity. Backward Euler convolution quadrature combined with Fourier pseudospectral discretization achieves mesh-uniform stability and positive-time first-order convergence. To accelerate the scheme, we utilize Cauchy--FFT weight recovery and adaptive blockwise sum-of-exponentials compression, which significantly reduces the cost of history evaluation under local error control. A logarithmic perturbation bound quantifies the effects of weight recovery, history compression, and inexact nonlinear solves, giving tolerance conditions that preserve the convergence order. Numerical experiments confirm the first-order accuracy and demonstrate that the accelerated method efficiently resolves long-time spatial patterns at a substantially reduced computational cost.
ER -