376 citations · 1.1k across the 19 of their papers we have counts for
20 papers
Error Profile for Discontinuous Galerkin Time Stepping of Parabolic PDEs
William McLean, Kassem Mustapha
We consider the time discretization of a linear parabolic problem by the discontinuous Galerkin (DG) method using piecewise polynomials of degree at most in , for …
Numerical Evaluation of Mittag-Leffler Functions
William McLean
The Mittag-Leffler function is computed via a quadrature approximation of a contour integral representation. We compare results for parabolic and hyperbolic contours, and give spec…
Uniform stability for a spatially-discrete, subdiffusive Fokker-Planck equation
William McLean, Kassem Mustapha
We prove stability estimates for the spatially discrete, Galerkin solution of a fractional Fokker-Planck equation, improving on previous results in several respects. Our main goal…
Implementation of high-order, discontinuous Galerkin time stepping for fractional diffusion problems
William McLean
The discontinuous Galerkin dG method provides a robust and flexible technique for the time integration of fractional diffusion problems. However, a practical implementation uses co…
A semidiscrete finite element approximation of a time-fractional Fokker-Planck equation with nonsmooth initial data
Kim Ngan Le, William McLean, Kassem Mustapha
We present a new stability and convergence analysis for the spatial discretization of a time-fractional Fokker--Planck equation in a convex polyhedral domain, using continuous, pie…
Existence, uniqueness and regularity of the solution of the time-fractional Fokker-Planck equation with general forcing
Kim-Ngan Le, William McLean, Martin Stynes
A time-fractional Fokker-Planck initial-boundary value problem is considered, with differential operator ,…
Regularity theory for time-fractional advection-diffusion-reaction equations
William McLean, Kassem Mustapha, Raed Ali +1
We investigate the behavior of the time derivatives of the solution to a linear time-fractional, advection-diffusion-reaction equation, allowing space- and time-dependent coefficie…
Well-posedness of time-fractional, advection-diffusion-reaction equations
William McLean, Kassem Mustapha, Raed Ali +1
We establish the well-posedness of an initial-boundary value problem for a general class of time-fractional, advection-diffusion-reaction equations, allowing space- and time-depend…
A discrete Grönwall inequality with application to numerical schemes for subdiffusion problems
Hong-lin Liao, William McLean, Jiwei Zhang
We consider a class of numerical approximations to the Caputo fractional derivative. Our assumptions permit the use of nonuniform time steps, such as is appropriate for accurately…
A second-order scheme with nonuniform time steps for a linear reaction-sudiffusion problem
Hong-lin Liao, William McLean, Jiwei Zhang
Stability and convergence of a time-weighted discrete scheme with nonuniform time steps are established for linear reaction-subdiffusion equations. The Caupto derivative is approxi…
Wider contours and adaptive contours
Shev MacNamara, William McLean, Kevin Burrage
Contour integrals in the complex plane are the basis of effective numerical methods for computing matrix functions, such as the matrix exponential and the Mittag-Leffler function.…
Fractional Euler Limits and Their Applications
Shev MacNamara, Bruce I Henry, William McLean
Generalisations of the classical Euler formula to the setting of fractional calculus are discussed. Compound interest and fractional compound interest serve as motivation. Connecti…
Exponential sum approximations for
William McLean
Given and , the function may be approximated for in a compact interval by a sum of terms of the form , with parameters and . On…
Finite element approximation of a time-fractional diffusion problem in a non-convex polygonal domain
Kim Ngan Le, William McLean, Bishnu Lamichhane
An initial-boundary value problem for the time-fractional diffusion equation is discretized in space using continuous piecewise-linear finite elements on a polygonal domain with a…
Numerical solution of the time-fractional Fokker-Planck equation with general forcing
Kim Ngan Le, William McLean, Kassem Mustapha
We study two schemes for a time-fractional Fokker-Planck equation with space- and time-dependent forcing in one space dimension. The first scheme is continuous in time and is discr…
Time-stepping error bounds for fractional diffusion problems with non-smooth initial data
William McLean, Kassem Mustapha
We apply the piecewise constant, discontinuous Galerkin method to discretize a fractional diffusion equation with respect to time. Using Laplace transform techniques, we show that…
Solving parabolic equations on the unit sphere via Laplace transforms and radial basis functions
Q. T. Le Gia, William McLean
We propose a method to construct numerical solutions of parabolic equations on the unit sphere. The time discretization uses Laplace transforms and quadrature. The spatial approxim…
Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations
Kassem Mustapha, William McLean
We consider an initial-boundary value problem for , that is, for a fractional diffusion () or wave () equation. A numerica…
Fast summation by interval clustering for an evolution equation with memory
William McLean
We solve a fractional diffusion equation using a piecewise-constant, discontinuous Galerkin method in time combined with a continuous, piecewise-linear finite element method in spa…
Iterative methods for shifted positive definite linear systems and time discretization of the heat equation
William McLean, Vidar Thomée
In earlier work we have studied a method for discretization in time of a parabolic problem which consists in representing the exact solution as an integral in the complex plane and…