A Time-Splitting Spectral Method for Three-Wave Interactions in Media with Competing Quadratic and Cubic Nonlinearities

This paper introduces an extension of the time-splitting spectral(TSSP)method for solving a general model of three-wave optical interactions,which typically arises from nonlinear optics,when the transmission media has competing quadratic and cubic nonlinearities.The key idea is to formulate the terms related to quadratic and cubic nonlinearities into a Hermitian matrix in a proper way,which allows us to develop an explicit and unconditionally stable numerical method for the problem.Furthermore,the method is spectral accurate in transverse coordinates and second-order accurate in propagation direction,is time reversible and time transverse invariant,and conserves the total wave energy(or power or the norm of the solutions)in discretized level.Numerical examples are presented to demonstrate the efficiency and high resolution of the method.Finally the method is applied to study dynamics and interactions between three-wave solitons and continuous waves in media with competing quadratic and cubic nonlinearities in one dimension(1D)and 2D.

On Time-Splitting Pseudospectral Discretization for Nonlinear Klein-Gordon Equation in Nonrelativistic Limit Regime

In this work,we are concerned with a time-splitting Fourier pseudospectral(TSFP)discretization for the Klein-Gordon(KG)equation,involving a dimensionless parameterε∈(0,1].In the nonrelativistic limit regime,the smallεproduces high oscillations in exact solutions with wavelength of O(ε^(−2))in time.The key idea behind the TSFP is to apply a time-splitting integrator to an equivalent first-order system in time,with both the nonlinear and linear subproblems exactly integrable in time and,respectively,Fourier frequency spaces.The method is fully explicit and time reversible.Moreover,we establish rigorously the optimal error bounds of a second-order TSFP for fixedε=O(1),thanks to an observation that the scheme coincides with a type of trigonometric integrator.As the second task,numerical studies are carried out,with special effortsmade to applying the TSFP in the nonrelativistic limit regime,which are geared towards understanding its temporal resolution capacity and meshing strategy for O(ε^(−2))-oscillatory solutions when 0<ε≪1.It suggests that the method has uniform spectral accuracy in space,and an asymptotic O(ε^(−2)D^(t2))temporal discretization error bound(Dt refers to time step).On the other hand,the temporal error bounds for most trigonometric integrators,such as the well-established Gautschi-type integrator in[6],are O(ε^(−4)D^(t2)).Thus,our method offers much better approximations than the Gautschi-type integrator in the highly oscillatory regime.These results,either rigorous or numerical,are valid for a splitting scheme applied to the classical relativistic NLS reformulation as well.

Theoretical analysis and numerical simulation of acoustic waves in gas hydrate-bearing sediments

Based on Carcione-Leclaire model,the time-splitting high-order staggered-grid finite-difference algorithm is proposed and constructed for understanding wave propagation mechanisms in gas hydrate-bearing sediments.Three compressional waves and two shear waves,as well as their energy distributions are investigated in detail.In particular,the influences of the friction coefficient between solid grains and gas hydrate and the viscosity of pore fluid on wave propagation are analyzed.The results show that our proposed numerical simulation algorithm proposed in this paper can effectively solve the problem of stiffness in the velocity-stress equations and suppress the grid dispersion,resulting in higher accuracy compared with the result of the Fourier pseudospectral method used by Carcione.The excitation mechanisms of the five wave modes are clearly revealed by the results of simulations.Besides,it is pointed that,the wave diffusion of the second kind of compressional and shear waves is influenced by the friction coefficient between solid grains and gas hydrate,while the diffusion of the third compressional wave is controlled by the fluid viscosity.Finally,two fluid-solid(gas-hydrate formation)models are constructed to study the mode conversion of various waves.The results show that the reflection,transmission,and transformation of various waves occur on the interface,forming a very complicated wave field,and the energy distribution of various converted waves in different phases is different.It is demonstrated from our studies that,the unconventional waves,such as the second and third kinds of compressional waves may be converted into conventional waves on an interface.These propagation mechanisms provide a concrete wave attenuation explanation in inhomogeneous media.

AN INTEGRATION METHOD WITH FITTING CUBIC SPLINE FUNCTIONS TO A NUMERICAL MODEL OF 2ND-ORDER SPACE-TIME DIFFERENTIAL REMAINDER——FOR AN IDEAL GLOBAL SIMULATION CASE WITH PRIMITIVE ATMOSPHERIC EQUATIONS

In this paper,the forecasting equations of a 2nd-order space-time differential remainder are deduced from the Navier-Stokes primitive equations and Eulerian operator by Taylor-series expansion.Here we introduce a cubic spline numerical model(Spline Model for short),which is with a quasi-Lagrangian time-split integration scheme of fitting cubic spline/bicubic surface to all physical variable fields in the atmospheric equations on spherical discrete latitude-longitude mesh.A new algorithm of"fitting cubic spline—time step integration—fitting cubic spline—……"is developed to determine their first-and2nd-order derivatives and their upstream points for time discrete integral to the governing equations in Spline Model.And the cubic spline function and its mathematical polarities are also discussed to understand the Spline Model’s mathematical foundation of numerical analysis.It is pointed out that the Spline Model has mathematical laws of"convergence"of the cubic spline functions contracting to the original functions as well as its 1st-order and 2nd-order derivatives.The"optimality"of the 2nd-order derivative of the cubic spline functions is optimal approximation to that of the original functions.In addition,a Hermite bicubic patch is equivalent to operate on a grid for a 2nd-order derivative variable field.Besides,the slopes and curvatures of a central difference are identified respectively,with a smoothing coefficient of 1/3,three-point smoothing of that of a cubic spline.Then the slopes and curvatures of a central difference are calculated from the smoothing coefficient 1/3 and three-point smoothing of that of a cubic spline,respectively.Furthermore,a global simulation case of adiabatic,non-frictional and"incompressible"model atmosphere is shown with the quasi-Lagrangian time integration by using a global Spline Model,whose initial condition comes from the NCEP reanalysis data,along with quasi-uniform latitude-longitude grids and the so-called"shallow atmosphere"Navier-Stokes primitive equations in the spherical coordinates.The Spline Model,which adopted the Navier-Stokes primitive equations and quasi-Lagrangian time-split integration scheme,provides an initial ideal case of global atmospheric circulation.In addition,considering the essentially non-linear atmospheric motions,the Spline Model could judge reasonably well simple points of any smoothed variable field according to its fitting spline curvatures that must conform to its physical interpretation.

An Efficient Numerical Method for the Quintic Complex Swift-Hohenberg Equation

In this paper,we present an efficient time-splitting Fourier spectral method for the quintic complex Swift-Hohenberg equation.Using the Strang time-splitting technique,we split the equation into linear part and nonlinear part.The linear part is solved with Fourier Pseudospectral method;the nonlinear part is solved analytically.We show that the method is easy to be applied and second-order in time and spectrally accurate in space.We apply the method to investigate soliton propagation,soliton interaction,and generation of stable moving pulses in one dimension and stable vortex solitons in two dimensions.

EFFICIENT AND ACCURATE NUMERICAL METHODS FOR LONG-WAVE SHORT-WAVE INTERACTION EQUATIONS IN THE SEMICLASSICAL LIMIT REGIME

This paper focuses on performance of several efficient and accurate numerical methods for the long-wave short-wave interaction equations in the semiclassical limit regime. The key features of the proposed methods are based on:(i) the utilization of the first-order or second-order time-splitting method to the nonlinear wave interaction equations;(ii) the ap-plication of Fourier pseudo-spectral method or compact finite difference approximation to the linear subproblem and the spatial derivatives;(iii) the adoption of the exact integration of the nonlinear subproblems and the ordinary differential equations in the phase space. The numerical methods under study are efficient, unconditionally stable and higher-order accurate, they are proved to preserve two invariants including the position density in L^1. Numerical results are reported for case studies with different types of initial data, these results verify the conservation laws in the discrete sense, show the dependence of the numerical solution on the time-step, mesh-size and dispersion parameter ε, and demonstrate the behavior of nonlinear dispersive waves in the semi-classical limit regime.

Numerical Integrators for Dispersion-Managed KdV Equation

In this paper,we consider the numerics of the dispersion-managed Kortewegde Vries(DM-KdV)equation for describingwave propagations in inhomogeneous media.The DM-KdV equation contains a variable dispersion map with discontinuity,which makes the solution non-smooth in time.We formally analyze the convergence order reduction problems of some popular numerical methods including finite difference and time-splitting for solving the DM-KdV equation,where a necessary constraint on the time step has been identified.Then,two exponential-type dispersionmap integrators up to second order accuracy are derived,which are efficiently incorporatedwith the Fourier pseudospectral discretization in space,and they can converge regardless the discontinuity and the step size.Numerical comparisons show the advantage of the proposed methods with the application to solitary wave dynamics and extension to the fast&strong dispersion-management regime.

STUDY OF NON-BOUSSINESQ EFFCET ON SEA SURFACE HEIGHT

A set of equations was derived for a non-Boussinesq ocean model in thispaper. A new time-splitting scheme was introduced which incorporates the 4th-order Runge-Kuttaexplicit scheme of low-frequency mode and an implicit scheme of high-frequency mode. With thismodel, potential temperature, salinity fields and sea surface height were calculated simultaneouslysuch that the numerical error of extrapolation of density field from the current time level to thenext one could be reduced while using the equation of mass conservation to determine sea surfaceheight. The non-Bouss-inesq effect on the density field and sea surface height was estimated bynumerical experiments in the final part of this paper.

Improving energetics in an ideal baroclinic instability case with a Physical Conserving Fidelity model

To improve the energetics in the life cycle of an ideal baroclinic instability case, we develop a Physical Conserving Fidelity model （F-model）, and we compare the simulations from the F-model to those of the traditional global spectral semi-implicit model （control model）. The results for spectral kinetic energy and its budget indicate different performances at smaller scales in the two models. A two-way energy flow emerges in the generation and rapid growth stage of the baroclinic disturbance in the F-model. However, only a downscale mechanism dominates in the control model. In the F-model, the meso- and smaller scales are energized initially, and then an active upscale nonlinear cascade occurs. Thus, disturbances at prior scales are forced by both downscale and upscale energy cascades and by conversion from potential energy. An analysis of the eddy kinetic energy budget also shows remarkable enhancement of the energy conversion rate in the F-model. As a result, characteristics of the ideal baroclinic wave are greatly improved in the F-model, in terms of both intensity and time of formation.