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Numerical solution of the GrossPitaevskii equation for BoseEinstein condensation
 J. Comput. Phys
"... We study the numerical solution of the timedependent GrossPitaevskii equation (GPE) describing a BoseEinstein condensate (BEC) at zero or very low temperature. In preparation for the numerics we scale the 3d GrossPitaevskii equation and obtain a fourparameter model. Identifying ‘extreme paramet ..."
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Cited by 111 (56 self)
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We study the numerical solution of the timedependent GrossPitaevskii equation (GPE) describing a BoseEinstein condensate (BEC) at zero or very low temperature. In preparation for the numerics we scale the 3d GrossPitaevskii equation and obtain a fourparameter model. Identifying ‘extreme parameter regimes’, the model is accessible to analytical perturbation theory, which justifies formal procedures well known in the physical literature: reduction to 2d and 1d GPEs, approximation of ground state solutions of the GPE and geometrical optics approximations. Then we use a timesplitting spectral method to discretize the timedependent GPE. Again, perturbation theory is used to understand the discretization scheme and to choose the spatial/temporal grid in dependence of the perturbation parameter. Extensive numerical examples in 1d, 2d and 3d for weak/strong interactions, defocusing/focusing nonlinearity, and zero/nonzero initial phase data are presented to demonstrate the power of the numerical method and to discuss the physics of BoseEinstein condensation.
Fourthorder time stepping for stiff PDEs
 SIAM J. SCI. COMPUT
, 2005
"... A modification of the exponential timedifferencing fourthorder Runge–Kutta method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to nondiagonal operators. A comparison is made ..."
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Cited by 94 (3 self)
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A modification of the exponential timedifferencing fourthorder Runge–Kutta method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to nondiagonal operators. A comparison is made of the performance of this modified exponential timedifferencing (ETD) scheme against the competing methods of implicitexplicit differencing, integrating factors, timesplitting, and Fornberg and Driscoll’s “sliders ” for the KdV, Kuramoto–Sivashinsky, Burgers, and Allen–Cahn equations in one space dimension. Implementation of the method is illustrated by short Matlab programs for two of the equations. It is found that for these applications with fixed time steps, the modified ETD scheme is the best.
Optimization of the splitstep Fourier method in modeling optical fiber communications systems
 J. Lightw. Technol
, 2003
"... Abstract—We studied the efficiency of different implementations of the splitstep Fourier method for solving the nonlinear Schrödinger equation that employ different stepsize selection criteria. We compared the performance of the different implementations for a variety of pulse formats and syste ..."
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Cited by 36 (4 self)
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Abstract—We studied the efficiency of different implementations of the splitstep Fourier method for solving the nonlinear Schrödinger equation that employ different stepsize selection criteria. We compared the performance of the different implementations for a variety of pulse formats and systems, including higher order solitons, collisions of soliton pulses, a singlechannel periodically stationary dispersionmanaged soliton system, and chirped return to zero systems with single and multiple channels. We introduce a globally thirdorder accurate splitstep scheme, in which a bound on the local error is used to select the step size. In many cases, this method is the most efficient when compared with commonly used stepsize selection criteria, and it is robust for a wide range of systems providing a systemindependent rule for choosing the step sizes. We find that a stepsize selection method based on limiting the nonlinear phase rotation of each step is not efficient for many opticalfiber transmission systems, although it works well for solitons. We also tested a method that uses a logarithmic stepsize distribution to bound the amount of spurious fourwave mixing. This method is as efficient as other secondorder schemes in the singlechannel dispersionmanaged soliton system, while it is not efficient in other cases including multichannel simulations. We find that in most cases, the simple approach in which the step size is held constant is the least efficient of all the methods. Finally, we implemented a method in which the step size is inversely proportional to the largest group velocity difference between channels. This scheme performs best in multichannel optical communications systems for the values of accuracy typically required in most transmission simulations. Index Terms—Adaptive algorithms, numerical analysis, optical fiber communication simulation, optical propagation, optical solitons, software peformance, splitstep Fourier method (SSFM), timefrequency analysis.
Fourth order timestepping for low dispersion Kortewegde Vries and nonlinear Schrödinger equation
 27 T.P. Liu, Development of singularities in the
, 2008
"... Abstract. Purely dispersive equations, such as the Kortewegde Vries and the nonlinear Schrödinger equations in the limit of small dispersion, have solutions to Cauchy problems with smooth initial data which develop a zone of rapid modulated oscillations in the region where the corresponding dispers ..."
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Cited by 30 (18 self)
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Abstract. Purely dispersive equations, such as the Kortewegde Vries and the nonlinear Schrödinger equations in the limit of small dispersion, have solutions to Cauchy problems with smooth initial data which develop a zone of rapid modulated oscillations in the region where the corresponding dispersionless equations have shocks or blowup. Fourth order timestepping in combination with spectral methods is beneficial to numerically resolve the steep gradients in the oscillatory region. We compare the performance of several fourth order methods for the Kortewegde Vries and the focusing and defocusing nonlinear Schrödinger equations in the small dispersion limit: an exponential timedifferencing fourthorder RungeKutta method as proposed by Cox and Matthews in the implementation by Kassam and Trefethen, integrating factors, timesplitting, Fornberg and Driscoll’s ‘sliders’, and an ODE solver in Matlab.
A composite Runge–Kutta method for the spectral solution of semilinear PDEs
 J. Comput. Phys
, 2002
"... A new composite Runge–Kutta (RK) method is proposed for semilinear partial differential equations such as Korteweg–de Vries, nonlinear Schrödinger, Kadomtsev– Petviashvili (KP), Kuramoto–Sivashinsky (KS), Cahn–Hilliard, and others having highorder derivatives in the linear term. The method uses Fou ..."
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Cited by 15 (0 self)
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A new composite Runge–Kutta (RK) method is proposed for semilinear partial differential equations such as Korteweg–de Vries, nonlinear Schrödinger, Kadomtsev– Petviashvili (KP), Kuramoto–Sivashinsky (KS), Cahn–Hilliard, and others having highorder derivatives in the linear term. The method uses Fourier collocation and the classical fourthorder RK method, except for the stiff linear modes, which are treated with a linearly implicit RK method. The composite RK method is simple to implement, indifferent to the distinction between dispersive and dissipative problems, and as efficient on test problems for KS and KP as any other generally applicable method. c ○ 2002 Elsevier Science (USA)
A split step approach for the 3D Maxwell’s equations
 J. Comput. Appl. Math
, 2003
"... Abstract. Splitstep procedures have previously been used successfully in a number of situations, e.g. for Hamiltonian systems, such as certain nonlinear wave equations. In this study, we note that one particular way to write the 3D Maxwell’s equations separates these into two parts, requiring in a ..."
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Cited by 13 (2 self)
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Abstract. Splitstep procedures have previously been used successfully in a number of situations, e.g. for Hamiltonian systems, such as certain nonlinear wave equations. In this study, we note that one particular way to write the 3D Maxwell’s equations separates these into two parts, requiring in all only the solution of six uncoupled 1D wave equations. The approach allows arbitrary orders of accuracy in both time and space, and features in many cases unconditional stability. 1.
Some unconditionally stable time stepping methods for the 3D Maxwell's equations
 Promises and Problems”, IEEE Software
, 2003
"... Almost all the di#culties that arise in the numerical solution of Maxwell's equations are due to material interfaces. In case that their geometrical features are much smaller than a typical wave length, one would like to use small space steps with large time steps. The first time stepping metho ..."
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Cited by 10 (3 self)
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Almost all the di#culties that arise in the numerical solution of Maxwell's equations are due to material interfaces. In case that their geometrical features are much smaller than a typical wave length, one would like to use small space steps with large time steps. The first time stepping method which combines a very low cost per time step with unconditional stability was the ADIFDTD method introduced in 1999. The present discussion starts with this method, and with an even more recent CrankNicolsonbased split step method with similar properties. We then explore how these methods can be made even more e#cient by combining them with techniques that increase their temporal accuracies. Key words: Maxwell's equations, ADIFDTD, CrankNicolson, split step, Richardson extrapolation, Deferred correction 2000 MSC: 65M70, 65M06, 78M2 1
Local spectral time splitting method for first and secondorder partial differential equations
, 2005
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Implicit–Explicit Multistep Methods for FastWave–SlowWave Problems
, 2011
"... Implicit–explicit (IMEX) linear multistep methods are examined with respect to their suitability for the integration of fastwave–slowwave problems in which the fast wave has relatively low amplitude and need not be accurately simulated. The widely used combination of trapezoidal implicit and leapf ..."
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Cited by 4 (0 self)
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Implicit–explicit (IMEX) linear multistep methods are examined with respect to their suitability for the integration of fastwave–slowwave problems in which the fast wave has relatively low amplitude and need not be accurately simulated. The widely used combination of trapezoidal implicit and leapfrog explicit differencing is compared to schemes based on Adams methods or on backward differencing. Two new families of methods are proposed that have good stability properties in fastwave–slowwave problems: one family is based on Adams methods and the other on backward schemes. Here the focus is primarily on four specific schemes drawn from these two families: a pair of Adams methods and a pair of backward methods that are either (i) optimized for thirdorder accuracy in the explicit component of the full IMEX scheme, or (ii) employ particularly good schemes for the implicit component. These new schemes are superior, in many respects, to the linear multistep IMEX schemes currently in use. The behavior of these schemes is compared theoretically in the context of the simple oscillation equation and also for the linearized equations governing stratified compressible flow. Several schemes are also tested in fully nonlinear simulations of gravity waves generated by a localized source in a shear flow. 1.
Error estimates for a fully discrete spectral scheme for a class of nonlinear, nonlocal dispersive wave equations,”
 Applied Numerical Mathematics,
, 2001
"... Abstract We analyze a fully discrete spectral method for the numerical solution of the initialand periodic boundaryvalue problem for two nonlinear, nonlocal, dispersive wave equations, the BenjaminOno and the Intermediate Long Wave equations. The equations are discretized in space by the standard ..."
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Cited by 4 (1 self)
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Abstract We analyze a fully discrete spectral method for the numerical solution of the initialand periodic boundaryvalue problem for two nonlinear, nonlocal, dispersive wave equations, the BenjaminOno and the Intermediate Long Wave equations. The equations are discretized in space by the standard FourierGalerkin spectral method and in time by the explicit leapfrog scheme. For the resulting fully discrete, conditionally stable scheme we prove an L 2 error bound of spectral accuracy in space and of secondorder accuracy in time.