Problems of the numerical solution of the Cauchy problem for a first-order differential-operator equation are discussed. A fundamental feature of the problem under study is that the equation includes a fractional power of the self-adjoint positive operator. In computational practice, rational approximations of the fractional power operator are widely used in various versions. The purpose of this work is to construct special approximations in time when the transition to a new level in time provided a set of standard problems for the operator and not for the fractional power operator. Stable splitting schemes with weights parameters are proposed for the additive representation of rational approximation for a fractional power operator. Possibilities of using similar time approximations for other problems are noted. The numerical solution of a two-dimensional non-stationary problem with a fractional power of the Laplace operator is also presented.
We establish an equivalence between two classes of methods for solving fractional diffusion problems, namely, Reduced Basis Methods (RBM) and Rational Krylov Methods (RKM). In particular, we demonstrate that several recently proposed RBMs for fractional diffusion can be interpreted as RKMs. This changed point of view allows us to give convergence proofs for some methods where none were previously available. We also propose a new RKM for fractional diffusion problems with poles chosen using the best rational approximation of the function $x^{-s}$ in the spectral interval of the spatial discreti…
In this paper, we propose a new choice of poles to define reliable rational Krylov methods. These methods are used for approximating function of positive definite matrices. In particular, the fractional power and the fractional resolvent are considered because of their importance in the numerical solution of fractional partial differential equations. The results of the numerical experiments we have carried out on some fractional models confirm that the proposed approach is promising.
In this paper, a non-uniform time-stepping convex-splitting numerical algorithm for solving the widely used time-fractional Cahn-Hilliard equation is introduced. The proposed numerical scheme employs the $L1^+$ formula for discretizing the time-fractional derivative and a second-order convex-splitting technique to deal with the non-linear term semi-implicitly. Then the pseudospectral method is utilized for spatial discretization. As a result, the fully discrete scheme has several advantages: second-order accurate in time, spectrally accurate in space, uniquely solvable, mass preserving, and un…
Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains
Southern University of Sciences and Technology（南方科技大学）；Nanyang Technological University（南洋理工大学）；Hong Kong Baptist University（香港浸会大学）；Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学研究所）
Fractional differential equations (FDEs) describe subdiffusion behavior of dynamical systems. Its non-local structure requires taking into account the whole evolution history during the time integration, which then possibly causes additional memory use to store the history, growing in time. An alternative to a quadrature for the history integral is to approximate the fractional kernel with the sum of exponentials, which is equivalent to considering the FDE solution as a sum of solutions to a system of ODEs. One possibility to construct this system is to approximate the Laplace spectrum of the …
In this paper we propose a method to compute the solution to the fractional diffusion equation on directed networks, which can be expressed in terms of the graph Laplacian $L$ as a product $f(L^T) \boldsymbol{b}$, where $f$ is a non-analytic function involving fractional powers and $\boldsymbol{b}$ is a given vector. The graph Laplacian is a singular matrix, causing Krylov methods for $f(L^T) \boldsymbol{b}$ to converge more slowly. In order to overcome this difficulty and achieve faster convergence, we use rational Krylov methods applied to a desingularized version of the graph Laplacian, obt…
In this paper, we study the numerical solutions of the multi-dimensional spatial fractional Allen-Cahn equations. After semi-discretization for the spatial fractional Riesz derivative, a system of nonlinear ordinary differential equations with Toeplitz structure is obtained. For the sake of reducing the computational complexity, a two-level Strang splitting method is proposed, where the Toeplitz matrix in the system is split into the sum of a circulant matrix and a skew-circulant matrix. Therefore, the proposed method can be quickly implemented by the fast Fourier transform, substituting to ca…
In this paper, we are concerned with a operator splitting scheme for linear fractional and fractional degenerate stochastic conservation laws driven by multiplicative Levy noise. More specifically, using a variant of classical Kruzkov's doubling of variable approach, we show that the approximate solutions generated by the splitting scheme converges to the unique stochastic entropy solution of the underlying problems.Finally, the convergence analysis is illustrated by several numerical examples.
We present a unified framework to efficiently approximate solutions to fractional diffusion problems of stationary and parabolic type. After discretization, we can take the point of view that the solution is obtained by a matrix-vector product of the form $f^{\boldsymbolτ}(L)\mathbf{b}$, where $L$ is the discretization matrix of the spatial operator, $\mathbf{b}$ a prescribed vector, and $f^{\boldsymbolτ}$ a parametric function, such as the fractional power or the Mittag-Leffler function. In the abstract framework of Stieltjes and complete Bernstein functions, to which the functions we are int…
Diagonal and normal with Toeplitz-block splitting iteration method for space fractional coupled nonlinear Schrödinger equations with repulsive nonlinearities
We develop a novel a posteriori error estimator for the $L^2$ error committed by the finite element discretization of the solution of the fractional Laplacian. Our a posteriori error estimator takes advantage of the semi-discretization scheme using rational approximations which allow to reformulate the fractional problem into a family of non-fractional parametric problems. The estimator involves applying the implicit Bank-Weiser error estimation strategy to each parametric non-fractional problem and reconstructing the fractional error through the same rational approximation used to compute the…
In this paper we consider one particular mathematical problem of this large area of fractional powers of self-adjoined elliptic operators, defined either by Dunford-Taylor-like integrals or by the representation through the spectrum of the elliptic operator. Due to the mathematical modeling of various non-local phenomena using such operators recently a number of numerical methods for solving equations involving operators of fractional order were introduced, studied, and tested. Here we consider the discrete counterpart of such problems obtained from finite difference or finite element approxim…
We consider a class of mathematical models describing multiphysics phenomena interacting through interfaces. On such interfaces, the traces of the fields lie (approximately) in the range of a weighted sum of two fractional differential operators. We use a rational function approximation to precondition such operators. We first demonstrate the robustness of the approximation for ordinary functions given by weighted sums of fractional exponents. Additionally, we present more realistic examples utilizing the proposed preconditioning techniques in interface coupling between Darcy and Stokes equati…
Here we study theoretically and compare experimentally an efficient method for solving systems of algebraic equations, where the matrix comes from the discretization of a fractional diffusion operator. More specifically, we focus on matrices obtained from finite difference or finite element approximation of second order elliptic problems in $\mathbb R^d$, $d=1,2,3$. The proposed methods are based on the best uniform rational approximation (BURA) $r_{α,k}(t)$ of $t^α$ on $[0,1]$. Here $r_{α,k}$ is a rational function of $t$ involving numerator and denominator polynomials of degree at most $k$. …
In this work, we consider a time-fractional Allen-Cahn equation, where the conventional first order time derivative is replaced by a Caputo fractional derivative with order $α\in(0,1)$. First, the well-posedness and (limited) smoothing property are systematically analyzed, by using the maximal $L^p$ regularity of fractional evolution equations and the fractional Grönwall's inequality. We also show the maximum principle like their conventional local-in-time counterpart. Precisely, the time-fractional equation preserves the property that the solution only takes value between the wells of the dou…
A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions
Department of Computer Sciences, Shahid Beheshti University（沙希德·贝赫什提大学计算机系）；Institute for Cognitive and Brain Sciences, Shahid Beheshti University（沙希德·贝赫什提大学认知与脑科学研究所）
Comparison analysis on two numerical methods for fractional diffusion problems based on rational approximations of $t^γ, \ 0 \le t \le 1$
Institute of Information and Communication Technologies, Bulgarian Academy of Sciences（保加利亚科学院信息与通信技术研究所以）；Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与计算机科学研究所）
Several applied problems are characterized by the need to numerically solve equations with an operator function (matrix function). In particular, in the last decade, mathematical models with a fractional power of an elliptic operator and numerical methods for their study have been actively discussed. Computational algorithms for such non-standard problems are based on approximations by the operator function. The most widespread are the approaches using various options for rational approximation. Also, we note the methods that relate to approximation by exponential sums. In this paper, the poss…
Splitting-based time integration approaches such as fractional steps, alternating direction implicit, operator splitting, and locally one-dimensional methods partition the system of interest into components and solve individual components implicitly in a cost-effective way. This work proposes a unified formulation of splitting time integration schemes in the framework of general-structure additive Runge-Kutta (GARK) methods. Specifically, we develop implicit-implicit (IMIM) GARK schemes, provide the order conditions and stability analysis for this class, and explain their application to partit…
In this paper, we propose a fourth order PDE model for image inpainting based on a variant of the famous Mumford-Shah (MS) image segmentation model. Convexity splitting is used to discrtised the time and we replace the Laplacian by its fractional counterpart in the time discretised scheme. Fourier spectral method is used for space discretization. Consistency, stability and convergence of the time discretised model has been carried out. The model is tested on some standard test images and compared them with the result of some models existed in the literature.
We propose a short-memory operator splitting scheme for solving the constant-Q wave equation, where the fractional stress-strain relation contains multiple Caputo fractional derivatives with order much smaller than 1. The key is to exploit its extension problem by converting the flat singular kernels into strongly localized ones, so that the major contribution of weakly singular integrals over a semi-infinite interval can be captured by a few Laguerre functions with proper asymptotic behavior. Despite its success in reducing both memory requirement and arithmetic complexity, we show that numer…
Global existence for vector valued fractional reaction-diffusion equations
Instituto de Matemática Luis Santaló, CONICET–UBA（路易斯·桑塔洛数学研究所，阿根廷国家科学研究委员会-布宜诺斯艾利斯大学）；Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires（布宜诺斯艾利斯大学理学院数学系）
