7,216 research outputs found
A k-space method for nonlinear wave propagation
A k-space method for nonlinear wave propagation in absorptive media is
presented. The Westervelt equation is first transferred into k-space via
Fourier transformation, and is solved by a modified wave-vector time-domain
scheme [Mast et al., IEEE Tran. Ultrason. Ferroelectr. Freq. Control 48,
341-354 (2001)]. The present approach is not limited to forward propagation or
parabolic approximation. One- and two-dimensional problems are investigated to
verify the method by comparing results to the finite element method. It is
found that, in order to obtain accurate results in homogeneous media, the grid
size can be as little as two points per wavelength, and for a moderately
nonlinear problem, the Courant-Friedrichs-Lewy number can be as small as 0.4.
As a result, the k-space method for nonlinear wave propagation is shown here to
be computationally more efficient than the conventional finite element method
or finite-difference time-domain method for the conditions studied here.
However, although the present method is highly accurate for weakly
inhomogeneous media, it is found to be less accurate for strongly inhomogeneous
media. A possible remedy to this limitation is discussed
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
New, efficient, and accurate high order derivative and dissipation operators satisfying summation by parts, and applications in three-dimensional multi-block evolutions
We construct new, efficient, and accurate high-order finite differencing
operators which satisfy summation by parts. Since these operators are not
uniquely defined, we consider several optimization criteria: minimizing the
bandwidth, the truncation error on the boundary points, the spectral radius, or
a combination of these. We examine in detail a set of operators that are up to
tenth order accurate in the interior, and we surprisingly find that a
combination of these optimizations can improve the operators' spectral radius
and accuracy by orders of magnitude in certain cases. We also construct
high-order dissipation operators that are compatible with these new finite
difference operators and which are semi-definite with respect to the
appropriate summation by parts scalar product. We test the stability and
accuracy of these new difference and dissipation operators by evolving a
three-dimensional scalar wave equation on a spherical domain consisting of
seven blocks, each discretized with a structured grid, and connected through
penalty boundary conditions.Comment: 16 pages, 9 figures. The files with the coefficients for the
derivative and dissipation operators can be accessed by downloading the
source code for the document. The files are located in the "coeffs"
subdirector
Fourier-Splitting methods for the dynamics of rotating Bose-Einstein condensates
We present a new method to propagate rotating Bose-Einstein condensates
subject to explicitly time-dependent trapping potentials. Using algebraic
techniques, we combine Magnus expansions and splitting methods to yield any
order methods for the multivariate and nonautonomous quadratic part of the
Hamiltonian that can be computed using only Fourier transforms at the cost of
solving a small system of polynomial equations. The resulting scheme solves the
challenging component of the (nonlinear) Hamiltonian and can be combined with
optimized splitting methods to yield efficient algorithms for rotating
Bose-Einstein condensates. The method is particularly efficient for potentials
that can be regarded as perturbed rotating and trapped condensates, e.g., for
small nonlinearities, since it retains the near-integrable structure of the
problem. For large nonlinearities, the method remains highly efficient if
higher order p > 2 is sought. Furthermore, we show how it can adapted to the
presence of dissipation terms. Numerical examples illustrate the performance of
the scheme.Comment: 15 pages, 4 figures, as submitted to journa
FFT-LB modeling of thermal liquid-vapor systems
We further develop a thermal LB model for multiphase flows. In the improved
model, we propose to use the FFT scheme to calculate both the convection term
and external force term. The usage of FFT scheme is detailed and analyzed. By
using the FFT algorithm spatiotemporal discretization errors are decreased
dramatically and the conservation of total energy is much better preserved. A
direct consequence of the improvement is that the unphysical spurious
velocities at the interfacial regions can be damped to neglectable scale.
Together with the better conservation of total energy, the more accurate flow
velocities lead to the more accurate temperature field which determines the
dynamical and final states of the system. With the new model, the phase diagram
of the liquid-vapor system obtained from simulation is more consistent with
that from theoretical calculation. Very sharp interfaces can be achieved. The
accuracy of simulation results are also verified by the Laplace law. The FFT
scheme can be easily applied to other models for multiphase flows.Comment: 34 pages, 21 figure
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