74 research outputs found
Alikhanov Legendre–Galerkin spectral method for the coupled nonlinear time-space fractional Ginzburg–Landau complex system
A finite difference/Galerkin spectral discretization for the temporal and spatial fractional coupled Ginzburg-Landau system is proposed and analyzed. The Alikhanov L2-1 sigma difference formula is utilized to discretize the time Caputo fractional derivative, while the Legendre-Galerkin spectral approximation is used to approximate the Riesz spatial fractional operator. The scheme is shown efficiently applicable with spectral accuracy in space and second-order in time. A discrete form of the fractional Gronwall inequality is applied to establish the error estimates of the approximate solution based on the discrete energy estimates technique. The key aspects of the implementation of the numerical continuation are complemented with some numerical experiments to confirm the theoretical claims
The impact of memory effect on space fractional strong quantum couplers with tunable decay behavior and its numerical simulation
The nontrivial behavior of wave packets in the space fractional coupled nonlinear Schrödinger equation has received considerable theoretical attention. The difficulty comes from the fact that the Riesz fractional derivative is inherently a prehistorical operator. In contrast, nonlinear Schrödinger equation with both time and space nonlocal operators, which is the cornerstone in the modeling of a new type of fractional quantum couplers, is still in high demand of attention. This paper is devoted to numerically study the propagation of solitons through a new type of quantum couplers which can be called time-space fractional quantum couplers. The numerical methodology is based on the finite-difference/Galerkin Legendre spectral method with an easy to implement numerical algorithm. The time-fractional derivative is considered to describe the decay behavior and the nonlocal memory of the model. We conduct numerical simulations to observe the performance of the tunable decay and the sharpness behavior of the time-space fractional strongly coupled nonlinear Schrödinger model as well as the performance of the numerical algorithm. Numerical simulations show that the time and space fractional-order operators control the decay behavior or the memory and the sharpness of the interface and undergo a seamless transition of the fractional-order parameters. © 2021, The Author(s).This study was supported financially by RFBR Grant (19-01-00019), the National Research Centre of Egypt (NRC) and Ghent university
Innovative Approaches to the Numerical Approximation of PDEs
This workshop was about the numerical solution of PDEs for which classical
approaches,
such as the finite element method, are not well suited or need further
(theoretical) underpinnings.
A prominent example of PDEs for which classical methods are not well
suited are PDEs posed in high space dimensions.
New results on low rank tensor approximation for those problems were
presented.
Other presentations dealt with regularity of PDEs, the numerical solution
of PDEs on surfaces,
PDEs of fractional order, numerical solvers for PDEs that converge with
exponential rates, and
the application of deep neural networks for solving PDEs
Finite element discretizations for variable-order fractional diffusion problems
We present a finite element scheme for fractional diffusion problems with
varying diffusivity and fractional order. We consider a symmetric integral form
of these nonlocal equations defined on general geometries and in arbitrary
bounded domains. A number of challenges are encountered when discretizing these
equations. The first comes from the heterogeneous kernel singularity in the
fractional integral operator. The second comes from the dense discrete operator
with its quadratic growth in memory footprint and arithmetic operations. An
additional challenge comes from the need to handle volume conditions-the
generalization of classical local boundary conditions to the nonlocal setting.
Satisfying these conditions requires that the effect of the whole domain,
including both the interior and exterior regions, can be computed on every
interior point in the discretization. Performed directly, this would result in
quadratic complexity. To address these challenges, we propose a strategy that
decomposes the stiffness matrix into three components. The first is a sparse
matrix that handles the singular near-field separately and is computed by
adapting singular quadrature techniques available for the homogeneous case to
the case of spatially variable order. The second component handles the
remaining smooth part of the near-field as well as the far field and is
approximated by a hierarchical matrix that maintains linear
complexity in storage and operations. The third component handles the effect of
the global mesh at every node and is written as a weighted mass matrix whose
density is computed by a fast-multipole type method. The resulting algorithm
has therefore overall linear space and time complexity. Analysis of the
consistency of the stiffness matrix is provided and numerical experiments are
conducted to illustrate the convergence and performance of the proposed
algorithm.Comment: 33 pages, 11 figure
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