arXiv subjects

William McLean

Publications and source records attributed to William McLean.

At least 19 recordsLinked to original sources

arXiv2024

Time-fractional diffusion equations with randomness, and efficient numerical estimations of expected values

In this work, we explore a time-fractional diffusion equation of order $\alpha \in (0,1)$ with a stochastic diffusivity parameter. We focus on efficient estimation of the expected values (considered as an infinite dimensional integral on the parametric space corresponding to the random coefficients) of linear functionals acting on the solution of our model problem. To estimate the expected value computationally, the infinite expansions of the random parameter need to be truncated. Then we approximate the high-dimensional integral over the random field using a high-order quasi-Monte Carlo method. This follows by approximating the deterministic solution over the space-time domain via a second-order accurate time-stepping scheme in combination with a spatial discretization by Galerkin finite elements. Under reasonable regularity assumptions on the given data, we show some regularity properties of the continuous solution and investigate the errors from estimating the expected value. We report on numerical experiments that complement the theoretical results.

math.NA↗

arXiv2024

An $\alpha$-robust and second-order accurate scheme for a subdiffusion equation

We investigate a second-order accurate time-stepping scheme for solving a time-fractional diffusion equation with a Caputo derivative of order~$\alpha \in (0,1)$. The basic idea of our scheme is based on local integration followed by linear interpolation. It reduces to the standard Crank--Nicolson scheme in the classical diffusion case, that is, as $\alpha\to 1$. Using a novel approach, we show that the proposed scheme is $\alpha$-robust and second-order accurate in the $L^2(L^2)$-norm, assuming a suitable time-graded mesh. For completeness, we use the Galerkin finite element method for the spatial discretization and discuss the error analysis under reasonable regularity assumptions on the given data. Some numerical results are presented at the end.

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arXiv2022

Error Profile for Discontinuous Galerkin Time Stepping of Parabolic PDEs

We consider the time discretization of a linear parabolic problem by the discontinuous Galerkin (DG) method using piecewise polynomials of degree at most $r-1$ in $t$, for $r\ge1$ and with maximum step size~$k$. It is well known that the spatial $L_2$-norm of the DG error is of optimal order $k^r$ globally in time, and is, for $r\ge2$, superconvergent of order $k^{2r-1}$ at the nodes. We show that on the $n$th subinterval $(t_{n-1},t_n)$, the dominant term in the DG error is proportional to the local right Radau polynomial of degree $r$. This error profile implies that the DG error is of order $k^{r+1}$ at the right-hand Gauss--Radau quadrature points in each interval. We show that the norm of the jump in the DG solution at the left end point $t_{n-1}$ provides an accurate \emph{a posteriori} estimate for the maximum error over the subinterval $(t_{n-1},t_n)$. Furthermore, a simple post-processing step yields a \emph{continuous} piecewise polynomial of degree $r$ with the optimal global convergence rate of order $k^{r+1}$. We illustrate these results with some numerical experiments.

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arXiv2022

Numerical Evaluation of Mittag-Leffler Functions

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 special attention to evaluation on the real line. The main point of difference with respect to similar approaches from the literature is the way that poles in the integrand are handled. Rational approximation of the Mittag-Leffler function on the negative real axis is also discussed.

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arXiv2020

Uniform stability for a spatially-discrete, subdiffusive Fokker-Planck equation

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 is to establish that the stability constants are bounded uniformly in the fractional diffusion exponent $α\in(0,1]$. In addition, we account for the presence of an inhomogeneous term and show a stability estimate for the gradient of the Galerkin solution. As a by-product, the proofs of error bounds for a standard finite element approximation are simplified.

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arXiv2020

Implementation of high-order, discontinuous Galerkin time stepping for fractional diffusion problems

The discontinuous Galerkin dG method provides a robust and flexible technique for the time integration of fractional diffusion problems. However, a practical implementation uses coefficients defined by integrals that are not easily evaluated. We describe specialised quadrature techniques that efficiently maintain the overall accuracy of the dG method. In addition, we observe in numerical experiments that known superconvergence properties of dG time stepping for classical diffusion problems carry over in a modified form to the fractional-order setting.

math.NA↗

arXiv2019

A semidiscrete finite element approximation of a time-fractional Fokker-Planck equation with nonsmooth initial data

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, piecewise-linear, finite elements. The forcing may depend on time as well as on the spatial variables, and the initial data may have low regularity. Our analysis uses a novel sequence of energy arguments in combination with a generalized Gronwall inequality. Although this theory covers only the spatial discretization, we present numerical experiments with a fully discrete scheme employing a very small time step, and observe results consistent with the predicted convergence behavior.

math.NA↗

arXiv2019

Existence, uniqueness and regularity of the solution of the time-fractional Fokker-Planck equation with general forcing

A time-fractional Fokker-Planck initial-boundary value problem is considered, with differential operator $u_t-\nabla\cdot(\partial_t^{1-α}κ_α\nabla u-\textbf{F}\partial_t^{1-α}u)$, where $0<α<1$. The forcing function $\textbf{F} = \textbf{F}(t,x)$, which is more difficult to analyse than the case $\textbf{F}=\textbf{F}(x)$ investigated previously by other authors. The spatial domain $Ω\subset\mathbb{R}^d$, where $d\ge 1$, has a smooth boundary. Existence, uniqueness and regularity of a mild solution $u$ is proved under the hypothesis that the initial data $u_0$ lies in $L^2(Ω)$. For $1/2<α<1$ and $u_0\in H^2(Ω)\cap H_0^1(Ω)$, it is shown that $u$ becomes a classical solution of the problem. Estimates of time derivatives of the classical solution are derived---these are known to be needed in numerical analyses of this problem.

math.AP↗

arXiv2019

Well-posedness of time-fractional, advection-diffusion-reaction equations

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-dependent coefficients as well as initial data that may have low regularity. Our analysis relies on novel energy methods in combination with a fractional Gronwall inequality and properties of fractional integrals.

math.AP↗

arXiv2019

Regularity theory for time-fractional advection-diffusion-reaction equations

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 coefficients as well as initial data that may have low regularity. Our focus is on proving estimates that are needed for the error analysis of numerical methods. The nonlocal nature of the fractional derivative creates substantial difficulties compared with the case of a classical parabolic PDE. In our analysis, we rely on novel energy methods in combination with a fractional Gronwall inequality and certain properties of fractional integrals.

math.AP↗

arXiv2018

A discrete Gr\"{o}nwall inequality with application to numerical schemes for subdiffusion problems

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 resolving the behavior of a solution whose derivatives are singular at~$t=0$. The main result is a type of fractional Gr\"{o}nwall inequality and we illustrate its use by outlining some stability and convergence estimates of schemes for fractional reaction-subdiffusion problems. This approach extends earlier work that used the familiar L1 approximation to the Caputo fractional derivative, and will facilitate the analysis of higher order and linearized fast schemes.

math.NA↗

arXiv2018

A second-order scheme with nonuniform time steps for a linear reaction-sudiffusion problem

Stability and convergence of a time-weighted discrete scheme with nonuniform time steps are established for linear reaction-subdiffusion equations. The Caupto derivative is approximated at an offset point by using linear and quadratic polynomial interpolation. Our analysis relies on two tools: a discrete fractional Gr\"{o}nwall inequality and the global consistency analysis. The new consistency analysis makes use of an interpolation error formula for quadratic polynomials, which leads to a convolution-type bound for the local truncation error. To exploit these two tools, some theoretical properties of the discrete kernels in the numerical Caputo formula are crucial and we investigate them intensively in the nonuniform setting. Taking the initial singularity of the solution into account, we obtain a sharp error estimate on nonuniform time meshes. The fully discrete scheme generates a second-order accurate solution on the graded mesh provided a proper grading parameter is employed. An example is presented to show the sharpness of our analysis.

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arXiv2017

Wider contours and adaptive contours

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. These methods provide successful ways to solve partial differential equations, such as convection--diffusion models. Part of the success of these methods comes from exploiting the freedom to choose the contour, by appealing to Cauchy's theorem. However, the pseudospectra of non-normal matrices or operators present a challenge for these methods: if the contour is too close to regions where the norm of the resolvent matrix is large, then the accuracy suffers. Important applications that involve non-normal matrices or operators include the Black--Scholes equation of finance, and Fokker--Planck equations for stochastic models arising in biology. Consequently, it is crucial to choose the contour carefully. As a remedy, we discuss choosing a contour that is wider than it might otherwise have been for a normal matrix or operator. We also suggest a semi-analytic approach to adapting the contour, in the form of a parabolic bound that is derived by estimating the field of values. To demonstrate the utility of the approaches that we advocate, we study three models in biology: a monomolecular reaction, a bimolecular reaction and a trimolecular reaction. Modelling and simulation of these reactions is done within the framework of Markov processes. We also consider non-Markov generalisations that have Mittag-Leffler waiting times instead of the usual exponential waiting times of a Markov process.

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arXiv2016

Fractional Euler Limits and Their Applications

Generalisations of the classical Euler formula to the setting of fractional calculus are discussed. Compound interest and fractional compound interest serve as motivation. Connections to fractional master equations are highlighted. An application to the Schlogl reactions with Mittag-Leffler waiting times is described.

math.CA↗

arXiv2016

Exponential sum approximations for $t^{-\beta}$

Given $\beta>0$ and $\delta>0$, the function $t^{-\beta}$ may be approximated for $t$ in a compact interval $[\delta,T]$ by a sum of terms of the form $we^{-at}$, with parameters $w>0$ and $a>0$. One such an approximation, studied by Beylkin and Monz\'on, is obtained by applying the trapezoidal rule to an integral representation of $t^{-\beta}$, after which Prony's method is applied to reduce the number of terms in the sum with essentially no loss of accuracy. We review this method, and then describe a similar approach based on an alternative integral representation. The main difference is that the new approach achieves much better results before the application of Prony's method; after applying Prony's method the performance of both is much the same.

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arXiv2016

Finite element approximation of a time-fractional diffusion problem in a non-convex polygonal domain

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 re-entrant corner. Known error bounds for the case of a convex polygon break down because the associated Poisson equation is no longer $H^2$-regular. In particular, the method is no longer second-order accurate if quasi-uniform triangulations are used. We prove that a suitable local mesh refinement about the re-entrant corner restores second-order convergence. In this way, we generalize known results for the classical heat equation due to Chatzipantelidis, Lazarov, Thom\'ee and Wahlbin.

math.NA↗

arXiv2015

Numerical solution of the time-fractional Fokker-Planck equation with general forcing

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 discretized in space using a piecewise-linear Galerkin finite element method. The second is continuous in space and employs a time-stepping procedure similar to the classical implicit Euler method. We show that the space discretization is second-order accurate in the spatial $L_2$-norm, uniformly in time, whereas the corresponding error for the time-stepping scheme is $O(k^\alpha)$ for a uniform time step $k$, where $\alpha\in(1/2,1)$ is the fractional diffusion parameter. In numerical experiments using a combined, fully-discrete method, we observe convergence behaviour consistent with these results.

math.NA↗

arXiv2014

Time-stepping error bounds for fractional diffusion problems with non-smooth initial data

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 the method is first order accurate at the \$n\$th time level \$t_n\$, but the error bound includes a factor \$t_n^{-1}\$ if we assume no smoothness of the initial data. We also show that for smoother initial data the growth in the error bound as \$t_n\$ decreases is milder, and in some cases absent altogether. Our error bounds generalize known results for the classical heat equation and are illustrated for a model problem.

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