2,376 research outputs found
High-order time-splitting Hermite and Fourier spectral methods
In this paper, we are concerned with the numerical solution of the time-dependent Gross-Pitaevskii Equation (GPE) involving a quasi-harmonic potential. Primarily, we consider discretisations that are based on spectral methods in space and higher-order exponential operator splitting methods in time. The resulting methods are favourable in view of accuracy and efficiency; moreover, geometric properties of the equation such as particle number and energy conservation are well captured. Regarding the spatial discretisation of the GPE, we consider two approaches. In the unbounded domain, we employ a spectral decomposition of the solution into Hermite basis functions: on the other hand. restricting the equation to a sufficiently large bounded domain, Fourier techniques are applicable. For the time integration of the GPE, we study various exponential operator splitting methods of convergence orders two, four, and six. Our main objective is to provide accuracy and efficiency comparisons of exponential operator splitting Fourier and Hermite pseudospectral methods for the time evolution of the GPE. Furthermore, we illustrate the effectiveness of higher-order time-splitting methods compared to standard integrators in a long-term integration
Fourier methods for the perturbed harmonic oscillator in linear and nonlinear Schr\"odinger equations
We consider the numerical integration of the Gross-Pitaevskii equation with a
potential trap given by a time-dependent harmonic potential or a small
perturbation thereof. Splitting methods are frequently used with Fourier
techniques since the system can be split into the kinetic and remaining part,
and each part can be solved efficiently using Fast Fourier Transforms. To split
the system into the quantum harmonic oscillator problem and the remaining part
allows to get higher accuracies in many cases, but it requires to change
between Hermite basis functions and the coordinate space, and this is not
efficient for time-dependent frequencies or strong nonlinearities. We show how
to build new methods which combine the advantages of using Fourier methods
while solving the timedependent harmonic oscillator exactly (or with a high
accuracy by using a Magnus integrator and an appropriate decomposition).Comment: 12 pages of RevTex4-1, 8 figures; substantially revised and extended
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Strong and auxiliary forms of the semi-Lagrangian method for incompressible flows
We present a review of the semi-Lagrangian method for advection-diusion and incompressible Navier-Stokes equations discretized with high-order methods. In particular, we compare the strong form where the departure points are computed directly via backwards integration with the auxiliary form where an auxiliary advection equation is solved instead; the latter is also referred to as Operator Integration Factor Splitting (OIFS) scheme. For intermediate size of time steps the auxiliary form is preferrable but for large time steps only the strong form is stable
Viriato: a Fourier-Hermite spectral code for strongly magnetised fluid-kinetic plasma dynamics
We report on the algorithms and numerical methods used in Viriato, a novel
fluid-kinetic code that solves two distinct sets of equations: (i) the Kinetic
Reduced Electron Heating Model (KREHM) equations [Zocco & Schekochihin, Phys.
Plasmas 18, 102309 (2011)] (which reduce to the standard Reduced-MHD equations
in the appropriate limit) and (ii) the kinetic reduced MHD (KRMHD) equations
[Schekochihin et al., Astrophys. J. Suppl. 182:310 (2009)]. Two main
applications of these equations are magnetised (Alfvenic) plasma turbulence and
magnetic reconnection. Viriato uses operator splitting (Strang or Godunov) to
separate the dynamics parallel and perpendicular to the ambient magnetic field
(assumed strong). Along the magnetic field, Viriato allows for either a
second-order accurate MacCormack method or, for higher accuracy, a
spectral-like scheme composed of the combination of a total variation
diminishing (TVD) third order Runge-Kutta method for the time derivative with a
7th order upwind scheme for the fluxes. Perpendicular to the field Viriato is
pseudo-spectral, and the time integration is performed by means of an iterative
predictor-corrector scheme. In addition, a distinctive feature of Viriato is
its spectral representation of the parallel velocity-space dependence, achieved
by means of a Hermite representation of the perturbed distribution function. A
series of linear and nonlinear benchmarks and tests are presented, including a
detailed analysis of 2D and 3D Orszag-Tang-type decaying turbulence, both in
fluid and kinetic regimes.Comment: 42 pages, 15 figures, submitted to J. Comp. Phy
Performance of affine-splitting pseudo-spectral methods for fractional complex Ginzburg-Landau equations
In this paper, we evaluate the performance of novel numerical methods for
solving one-dimensional nonlinear fractional dispersive and dissipative
evolution equations. The methods are based on affine combinations of
time-splitting integrators and pseudo-spectral discretizations using Hermite
and Fourier expansions. We show the effectiveness of the proposed methods by
numerically computing the dynamics of soliton solutions of the the standard and
fractional variants of the nonlinear Schr\"odinger equation (NLSE) and the
complex Ginzburg-Landau equation (CGLE), and by comparing the results with
those obtained by standard splitting integrators. An exhaustive numerical
investigation shows that the new technique is competitive with traditional
composition-splitting schemes for the case of Hamiltonian problems both in
terms accuracy and computational cost. Moreover, it is applicable
straightforwardly to irreversible models, outperforming high-order symplectic
integrators which could become unstable due to their need of negative time
steps. Finally, we discuss potential improvements of the numerical methods
aimed to increase their efficiency, and possible applications to the
investigation of dissipative solitons that arise in nonlinear optical systems
of contemporary interest. Overall, our method offers a promising alternative
for solving a wide range of evolutionary partial differential equations.Comment: 31 pages, 12 figure
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