5,628 research outputs found
A multiscale collocation method for fractional differential problems
We introduce a multiscale collocation method to numerically solve differential problems involving both ordinary and fractional
derivatives of high order. The proposed method uses multiresolution analyses (MRA) as approximating spaces and takes advantage
of a finite difference formula that allows us to express both ordinary and fractional derivatives of the approximating function in a closed form. Thus, the method is easy to implement, accurate and efficient. The convergence and the stability of the multiscale
collocation method are proved and some numerical results are shown.We introduce a multiscale collocation method to numerically solve differential problems involving both ordinary and fractional
derivatives of high order. The proposed method uses multiresolution analyses (MRA) as approximating spaces and takes advantage
of a finite difference formula that allows us to express both ordinary and fractional derivatives of the approximating function in a closed form. Thus, the method is easy to implement, accurate and efficient. The convergence and the stability of the multiscale
collocation method are proved and some numerical results are shown
Solitary waves in the Nonlinear Dirac Equation
In the present work, we consider the existence, stability, and dynamics of
solitary waves in the nonlinear Dirac equation. We start by introducing the
Soler model of self-interacting spinors, and discuss its localized waveforms in
one, two, and three spatial dimensions and the equations they satisfy. We
present the associated explicit solutions in one dimension and numerically
obtain their analogues in higher dimensions. The stability is subsequently
discussed from a theoretical perspective and then complemented with numerical
computations. Finally, the dynamics of the solutions is explored and compared
to its non-relativistic analogue, which is the nonlinear Schr{\"o}dinger
equation. A few special topics are also explored, including the discrete
variant of the nonlinear Dirac equation and its solitary wave properties, as
well as the PT-symmetric variant of the model
Multipole-Preserving Quadratures for Discretization of Functions in Real-Space Electronic Structure Calculations
Discretizing an analytic function on a uniform real-space grid is often done
via a straightforward collocation method. This is ubiquitous in all areas of
computational physics and quantum chemistry. An example in Density Functional
Theory (DFT) is given by the external potential or the pseudo-potential
describing the interaction between ions and electrons. The accuracy of the
collocation method used is therefore very important for the reliability of
subsequent treatments like self-consistent field solutions of the electronic
structure problems. By construction, the collocation method introduces
numerical artifacts typical of real-space treatments, like the so-called
egg-box error, that may spoil the numerical stability of the description when
the real-space grid is too coarse. As the external potential is an input of the
problem, even a highly precise computational treatment cannot cope this
inconvenience. We present in this paper a new quadrature scheme that is able to
exactly preserve the moments of a given analytic function even for large grid
spacings, while reconciling with the traditional collocation method when the
grid spacing is small enough. In the context of real-space electronic structure
calculations, we show that this method improves considerably the stability of
the results for large grid spacings, opening the path towards reliable
low-accuracy DFT calculations with reduced number of degrees of freedom.Comment: 20 pages, 7 figure
A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces
In this paper we present a high-order kernel method for numerically solving
diffusion and reaction-diffusion partial differential equations (PDEs) on
smooth, closed surfaces embedded in . For two-dimensional
surfaces embedded in , these types of problems have received
growing interest in biology, chemistry, and computer graphics to model such
things as diffusion of chemicals on biological cells or membranes, pattern
formations in biology, nonlinear chemical oscillators in excitable media, and
texture mappings. Our kernel method is based on radial basis functions (RBFs)
and uses a semi-discrete approach (or the method-of-lines) in which the surface
derivative operators that appear in the PDEs are approximated using
collocation. The method only requires nodes at "scattered" locations on the
surface and the corresponding normal vectors to the surface. Additionally, it
does not rely on any surface-based metrics and avoids any intrinsic coordinate
systems, and thus does not suffer from any coordinate distortions or
singularities. We provide error estimates for the kernel-based approximate
surface derivative operators and numerically study the accuracy and stability
of the method. Applications to different non-linear systems of PDEs that arise
in biology and chemistry are also presented
Optimal B-spline bases for the numerical solution of fractional differential problems
Efficient numerical methods to solve fractional differential problems are particularly required in order to approximate accurately the nonlocal behavior of the fractional derivative. The aim of the paper is to show how optimal B-spline bases allow us to construct accurate numerical methods that have a low computational cost. First of all, we describe in detail how to construct optimal B-spline bases on bounded intervals and recall their main properties. Then, we give the analytical expression of their derivatives of fractional order and use these bases in the numerical solution of fractional differential problems. Some numerical tests showing the good performances of the bases in solving a time-fractional diffusion problem by a collocation-Galerkin method are also displayed
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