49 research outputs found
Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains
Many PDEs involving fractional Laplacian are naturally set in unbounded
domains with underlying solutions decay very slowly, subject to certain power
laws. Their numerical solutions are under-explored. This paper aims at
developing accurate spectral methods using rational basis (or modified mapped
Gegenbauer functions) for such models in unbounded domains. The main building
block of the spectral algorithms is the explicit representations for the
Fourier transform and fractional Laplacian of the rational basis, derived from
some useful integral identites related to modified Bessel functions. With these
at our disposal, we can construct rational spectral-Galerkin and direct
collocation schemes by pre-computing the associated fractional differentiation
matrices. We obtain optimal error estimates of rational spectral approximation
in the fractional Sobolev spaces, and analyze the optimal convergence of the
proposed Galerkin scheme. We also provide ample numerical results to show that
the rational method outperforms the Hermite function approach
Fast Fourier-like Mapped Chebyshev Spectral-Galerkin Methods for PDEs with Integral Fractional Laplacian in Unbounded Domains
In this paper, we propose a fast spectral-Galerkin method for solving PDEs
involving integral fractional Laplacian in , which is built upon
two essential components: (i) the Dunford-Taylor formulation of the fractional
Laplacian; and (ii) Fourier-like bi-orthogonal mapped Chebyshev functions
(MCFs) as basis functions. As a result, the fractional Laplacian can be fully
diagonalised, and the complexity of solving an elliptic fractional PDE is
quasi-optimal, i.e., with being the number of modes in
each spatial direction. Ample numerical tests for various decaying exact
solutions show that the convergence of the fast solver perfectly matches the
order of theoretical error estimates. With a suitable time-discretization, the
fast solver can be directly applied to a large class of nonlinear fractional
PDEs. As an example, we solve the fractional nonlinear Schr{\"o}dinger equation
by using the fourth-order time-splitting method together with the proposed
MCF-spectral-Galerkin method.Comment: This article has a total of 24 pages and including 22 figure
A finite-volume scheme for fractional diffusion on bounded domains
We propose a new fractional Laplacian for bounded domains, expressed as a conservation law and thus particularly suited to finite-volume schemes. Our approach permits the direct prescription of no-flux boundary conditions. We first show the well-posedness theory for the fractional heat equation. We also develop a numerical scheme, which correctly captures the action of the fractional Laplacian and its anomalous diffusion effect. We benchmark numerical solutions for the Lévy–Fokker–Planck equation against known analytical solutions. We conclude by numerically exploring properties of these equations with respect to their stationary states and long-time asymptotics
MultiShape: A Spectral Element Method, with Applications to Dynamic Density Functional Theory and PDE-Constrained Optimization
A numerical framework is developed to solve various types of PDEs on
complicated domains, including steady and time-dependent, non-linear and
non-local PDEs, with different boundary conditions that can also include
non-linear and non-local terms. This numerical framework, called MultiShape, is
a class in Matlab, and the software is open source. We demonstrate that
MultiShape is compatible with other numerical methods, such as
differential--algebraic equation solvers and optimization algorithms. The
numerical implementation is designed to be user-friendly, with most of the
set-up and computations done automatically by MultiShape and with intuitive
operator definition, notation, and user-interface. Validation tests are
presented, before we introduce three examples motivated by applications in
Dynamic Density Functional Theory and PDE-constrained optimization,
illustrating the versatility of the method
Towards an Efficient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains
We explore the connection between fractional order partial differential
equations in two or more spatial dimensions with boundary integral operators to
develop techniques that enable one to efficiently tackle the integral
fractional Laplacian. In particular, we develop techniques for the treatment of
the dense stiffness matrix including the computation of the entries, the
efficient assembly and storage of a sparse approximation and the efficient
solution of the resulting equations. The main idea consists of generalising
proven techniques for the treatment of boundary integral equations to general
fractional orders. Importantly, the approximation does not make any strong
assumptions on the shape of the underlying domain and does not rely on any
special structure of the matrix that could be exploited by fast transforms. We
demonstrate the flexibility and performance of this approach in a couple of
two-dimensional numerical examples