# Generalized convolution quadrature for the fractional integral and fractional diffusion equations

@article{Guo2022GeneralizedCQ, title={Generalized convolution quadrature for the fractional integral and fractional diffusion equations}, author={Jing Guo and Mar{\'i}a L{\'o}pez-Fern{\'a}ndez}, journal={ArXiv}, year={2022}, volume={abs/2211.13862} }

We consider the application of the generalized Convolution Quadrature (gCQ) of the first order to approximate fractional integrals and associated fractional diffusion equations. The gCQ is a generalization of Lubich's Convolution Quadrature (CQ) which allows for variable steps. In this paper we analyze the application of the gCQ to fractional integrals, with a focus on the low regularity case. It is well known that in this situation the original CQ presents an order reduction close to the…

## References

SHOWING 1-10 OF 19 REFERENCES

### Efficient high order algorithms for fractional integrals and fractional differential equations

- Mathematics, Computer ScienceNumerische Mathematik
- 2019

The error analysis of the full space-time discretization obtained by coupling the FEM method in space with Runge–Kutta based convolution quadrature in time is given.

### Generalized convolution quadrature with variable time stepping. Part II

- Mathematics, Computer Science
- 2015

### Convolution quadrature and discretized operational calculus. I

- Mathematics
- 1988

SummaryNumerical methods are derived for problems in integral equations (Volterra, Wiener-Hopf equations) and numerical integration (singular integrands, multiple time-scale convolution). The basic…

### Error Analysis of a Finite Difference Method on Graded Meshes for a Time-Fractional Diffusion Equation

- MathematicsSIAM J. Numer. Anal.
- 2017

The final convergence result shows clearly how the regularity of the solution and the grading of the mesh affect the order of convergence of the difference scheme, so one can choose an optimal mesh grading.

### Generalized convolution quadrature with variable time stepping

- Mathematics
- 2013

In this paper, we will present a generalized convolution quadrature for solving linear parabolic and hyperbolic evolution equations. The original convolution quadrature method by Lubich works very…

### An L1 approximation for a fractional reaction-diffusion equation, a second-order error analysis over time-graded meshes

- MathematicsSIAM J. Numer. Anal.
- 2020

This is the first paper to show that the L1 scheme for the model problem under consideration is second-order accurate (sharp error estimate) over nonuniform time-steps over Riemann--Liouville time-fractional derivative.

### Sharp Error Estimate of the Nonuniform L1 Formula for Linear Reaction-Subdiffusion Equations

- MathematicsSIAM J. Numer. Anal.
- 2018

A sharp error estimate reflecting the regularity of solution is obtained for a simple L1 scheme with the help of discrete fractional Gronwall inequality and global consistency error analysis.

### RUNGE-KUTTA METHODS FOR PARABOLIC EQUATIONS AND CONVOLUTION QUADRATURE

- Mathematics, Computer Science
- 1993

We study the approximation properties of Runge-Kutta time discretizations of linear and semilinear parabolic equations, including incompressible Navier-Stokes equations. We derive asymptotically…

### Generalized convolution quadrature based on Runge-Kutta methods

- Mathematics, Computer ScienceNumerische Mathematik
- 2016

The Runge-Kutta generalized convolution quadrature with variable time stepping for the numerical solution of convolution equations for time and space-time problems is developed and the corresponding stability and convergence analysis is presented.

### An error analysis of Runge–Kutta convolution quadrature

- Mathematics
- 2011

An error analysis is given for convolution quadratures based on strongly A-stable Runge–Kutta methods, for the non-sectorial case of a convolution kernel with a Laplace transform that is polynomially…