Multi-symplectic Runge-Kutta methods for Landau-Ginzburg-Higgs equation

Nonlinear wave equations have been extensively investigated in the last sev- eral decades. The Landau-Ginzburg-Higgs equation, a typical nonlinear wave equation, is studied in this paper based on the multi-symplectic theory in the Hamilton space. The multi-symplectic Runge-Kutta method is reviewed, and a semi-implicit scheme with certain discrete conservation laws is constructed to solve the first-order partial differential equations （PDEs） derived from the Landau-Ginzburg-Higgs equation. The numerical re- sults for the soliton solution of the Landau-Ginzburg-Higgs equation are reported, showing that the multi-symplectic Runge-Kutta method is an efficient algorithm with excellent long-time numerical behaviors.

High order symplectic conservative perturbation method for time-varying Hamiltonian system

This paper presents a high order symplectic con- servative perturbation method for linear time-varying Hamil- tonian system. Firstly, the dynamic equation of Hamilto- nian system is gradually changed into a high order pertur- bation equation, which is solved approximately by resolv- ing the Hamiltonian coefficient matrix into a ＂major compo- nent＂ and a ＂high order small quantity＂ and using perturba- tion transformation technique, then the solution to the orig- inal equation of Hamiltonian system is determined through a series of inverse transform. Because the transfer matrix determined by the method in this paper is the product of a series of exponential matrixes, the transfer matrix is a sym- plectic matrix; furthermore, the exponential matrices can be calculated accurately by the precise time integration method, so the method presented in this paper has fine accuracy, ef- ficiency and stability. The examples show that the proposed method can also give good results even though a large time step is selected, and with the increase of the perturbation or- der, the perturbation solutions tend to exact solutions rapidly.

A High-Order Conservative Numerical Method for Gross-Pitaevskii Equation with Time-Varying Coefficients in Modeling BEC

We propose a high-order conservative method for the nonlinear Sehodinger/Gross-Pitaevskii equation with time- varying coefficients in modeling Bose Einstein condensation （BEC）. This scheme combined with the sixth-order compact finite difference method and the fourth-order average vector field method, finely describes the condensate wave function and physical characteristics in some small potential wells. Numerical experiments are presented to demonstrate that our numerical scheme is efficient by the comparison with the Fourier pseudo-spectral method. Moreover, it preserves several conservation laws well and even exactly under some specific conditions.

A three-dimensional Eulerian method for the numerical simulation of high-velocity impact problems

In the present paper, a three-dimensional （3D） Eulerian technique for the 3D numerical simulation of high-velocity impact problems is proposed. In the Eulerian framework, a complete 3D conservation element and solution element scheme for conservative hyperbolic governing equations with source terms is given. A modified ghost fluid method is proposed for the treatment of the boundary conditions. Numerical simulations of the Taylor bar problem and the ricochet phenomenon of a sphere impacting a plate target at an angle of 60~ are carried out. The numerical results are in good agreement with the corresponding experimental observations. It is proved that our computational technique is feasible for analyzing 3D high-velocity impact problems.

Numerical simulation of oxide nanoparticle growth characteristics under the gas detonation chemical reaction by space-time conservation element-solution element method

Under harsh conditions （such as high temperature, high pressure, and millisecond lifetime chemical reaction）, a long-standing challenge remains to accurately predict the growth characteristics of nanosize spherical particles and to determine the rapid chemical reaction flow field characteristics, The growth characteristics of similar spherical oxide nanoparticles are further studied by successfully introducing the space-time conservation element-solution element （CE/SE） algorithm with the monodisperse Kruis model. This approach overcomes the nanosize particle rapid growth limit set and successfully captures the characteristics of the rapid gaseous chemical reaction process. The results show that this approach quantitatively captures the characteristics of the rapid chemical reaction, nanosize particle growth and size distribution. To reveal the growth mechanism for numerous types of oxide nanoparticles, it is very important to choose a rational numerical method and particle physics model.

Progress in satellite gravity recovery from implemented CHAMP,GRACE and GOCE and future GRACE Follow-On missions

Firstly, the Earth＇s gravitational field from the past Challenging Minisatellite Payload （CHAMP） mission is determined using the energy conservation principle, the combined error model of the cumulative geoid height influenced by three instrument errors from the current Gravity Recovery and Climate Experiment （GRACE） and future GRACE Follow-On missions is established based on the semi-analytical method, and the Earth＇s gravitational field from the executed Gravity Field and Steady-State Ocean Circulation Explorer （GOCE） mission is recovered by the space-time-wise approach. Secondly, the cumulative geoid height errors are 1.727 × 10^-1 m, 1.839 × 10^-1 m and 9.025 × 10^ -2 m at degrees 70,120 and 250 from the implemented three-stage satellite gravity missions consisting of CHAMP, GRACE and GOCE, which preferably accord with those from the existing earth gravity field models involving EIGEN-CHAMP03S, EICEN-GRACE02S and GO_CONS GCF 2 DIR R1. The cumulative geoid height error is 6.847 × 10 ^-2 m at degree 250 from the future GRACE Follow-On mission. Finally, the complementarity among the four-stage satellite gravity missions including CHAMP, GRACE, GOCE and GRACE Follow-On is demonstrated contrastively.

A Positivity-preserving Conservative Semi-Lagrangian Multi-moment Global Transport Model on the Cubed Sphere

A positivity-preserving conservative semi-Lagrangian transport model by multi-moment finite volume method has been developed on the cubed-sphere grid.Two kinds of moments(i.e.,point values(PV moment) at cell interfaces and volume integrated average(VIA moment) value) are defined within a single cell.The PV moment is updated by a conventional semi-Lagrangian method,while the VIA moment is cast by the flux form formulation to assure the exact numerical conservation.Different from the spatial approximation used in the CSL2(conservative semi-Lagrangian scheme with second order polynomial function) scheme,a monotonic rational function which can effectively remove non-physical oscillations is reconstructed within a single cell by the PV moments and VIA moment.To achieve exactly positive-definite preserving,two kinds of corrections are made on the original conservative semi-Lagrangian with rational function(CSLR)scheme.The resulting scheme is inherently conservative,non-negative,and allows a Courant number larger than one.Moreover,the spatial reconstruction can be performed within a single cell,which is very efficient and economical for practical implementation.In addition,a dimension-splitting approach coupled with multi-moment finite volume scheme is adopted on cubed-sphere geometry,which benefitsthe implementation of the 1 D CSLR solver with large Courant number.The proposed model is evaluated by several widely used benchmark tests on cubed-sphere geometry.Numerical results show that the proposed transport model can effectively remove nonphysical oscillations and preserve the numerical nonnegativity,and it has the potential to transport the tracers accurately in a real atmospheric model.

High-Order and High Accurate CFD Methods and Their Applications for Complex Grid Problems

The purpose of this article is to summarize our recent progress in high-order and high accurate CFD methods for flow problems with complex grids as well as to discuss the engineering prospects in using these methods.Despite the rapid development of high-order algorithms in CFD,the applications of high-order and high accurate methods on complex configurations are still limited.One of the main reasons which hinder the widely applications of thesemethods is the complexity of grids.Many aspects which can be neglected for low-order schemes must be treated carefully for high-order ones when the configurations are complex.In order to implement highorder finite difference schemes on complex multi-block grids,the geometric conservation lawand block-interface conditions are discussed.A conservativemetricmethod is applied to calculate the grid derivatives,and a characteristic-based interface condition is employed to fulfil high-order multi-block computing.The fifth-order WCNS-E-5 proposed by Deng[9,10]is applied to simulate flows with complex grids,including a double-delta wing,a transonic airplane configuration,and a hypersonic X-38 configuration.The results in this paper and the references show pleasant prospects in engineering-oriented applications of high-order schemes.

A Posteriori Error Estimates for Conservative Local Discontinuous Galerkin Methods for the Generalized Korteweg-de Vries Equation

We construct and analyze conservative local discontinuous Galerkin(LDG)methods for the Generalized Korteweg-de-Vries equation.LDG methods are designed by writing the equation as a system and performing separate approximations to the spatial derivatives.The main focus is on the development of conservative methods which can preserve discrete versions of the first two invariants of the continuous solution,and a posteriori error estimates for a fully discrete approximation that is based on the idea of dispersive reconstruction.Numerical experiments are provided to verify the theoretical estimates.

Investigation of the high-spin rotational properties of the proton emitter ^113Cs using a particle-number conserving method

The recently observed two high-spin rotational bands in the proton emitter ^113Cs are investigated using the cranked shell model with pairing correlations treated by a particle-number conserving method, in which the Pauli blocking effects are taken into account exactly. By using the configuration assignments of band 1 [π3/2^＋[422]（g7/2）, α =-1/2] and band 2 [π1/2^＋[420]（d5/2）, α=1/2], the experimental moments of inertia and quasiparticle alignments can be reproduced much better by the present calculations than those using the configuration assginment of π1/2^-[550]（h11/2）, which in turn may support these configuration assignments. Furthermore, by analyzing the occupation probability nμ of each cranked Nilsson level near the Fermi surface and the contribution of each orbital to the angular momentum alignments, the backbending mechanism of these two bands is also investigated.

Robust Conservative Level SetMethod for 3D Mixed-Element Meshes —Application to LES of Primary Liquid-Sheet Breakup

In this article we detail the methodology developed to construct an efficient interface description technique—the robust conservative level set(RCLS)—to simulate multiphase flows on mixed-element unstructured meshes while conserving mass to machine accuracy.The approach is tailored specifically for industry as the three-dimensional unstructured approach allows for the treatment of very complex geometries.In addition,special care has been taken to optimise the trade-off between accuracy and computational cost while maintaining the robustness of the numerical method.This was achieved by solving the transport equations for the liquid volume fraction using a WENO scheme for polyhedral meshes and by adding a flux-limiter algorithm.The performance of the resulting method has been compared against established multiphase numerical methods and its ability to capture the physics of multiphase flows is demonstrated on a range of relevant test cases.Finally,the RCLS method has been applied to the simulation of the primary breakup of a flat liquid sheet of kerosene in co-flowing high-pressure gas.This quasi-DNS/LES computation was performed at relevant aero-engine conditions on a three-dimensional mixed-element unstructured mesh.The numerical results have been validated qualitatively against theoretical predictions and experimental data.In particular,the expected breakup regime was observed in the simulation results.Finally,the computation reproduced faithfully the breakup length predicted by a correlation based on experimental data.This constitutes a first step towards a quantitative validation.

A Space-Time Conservative Method for Hyperbolic Systems with Stiff and Non Stiff Source Terms

In this article we propose a higher-order space-time conservative method for hyperbolic systems with stiff and non stiff source terms as well as relaxation systems.We call the scheme a slope propagation(SP)method.It is an extension of our scheme derived for homogeneous hyperbolic systems[1].In the present inhomogeneous systems the relaxation time may vary from order of one to a very small value.These small values make the relaxation term stronger and highly stiff.In such situations underresolved numerical schemes may produce spurious numerical results.However,our present scheme has the capability to correctly capture the behavior of the physical phenomena with high order accuracy even if the initial layer and the small relaxation time are not numerically resolved.The scheme treats the space and time in a unified manner.The flow variables and their slopes are the basic unknowns in the scheme.The source term is treated by its volumetric integration over the space-time control volume and is a direct part of the overall space-time flux balance.We use two approaches for the slope calculations of the flow variables,the first one results directly from the flux balance over the control volumes,while in the second one we use a finite difference approach.The main features of the scheme are its simplicity,its Jacobian-free and Riemann solver-free recipe,as well as its efficiency and high of order accuracy.In particular we show that the scheme has a discrete analog of the continuous asymptotic limit.We have implemented our scheme for various test models available in the literature such as the Broadwell model,the extended thermodynamics equations,the shallow water equations,traffic flow and the Euler equations with heat transfer.The numerical results validate the accuracy,versatility and robustness of the present scheme.

A Conservative Local Discontinuous Galerkin Method for the Schrödinger-KdV System

In this paper we develop a conservative local discontinuous Galerkin(LDG)method for the Schrödinger-Korteweg-de Vries(Sch-KdV)system,which arises in various physical contexts as a model for the interaction of long and short nonlinear waves.Conservative quantities in the discrete version of the number of plasmons,energy of the oscillations and the number of particles are proved for the LDG scheme of the Sch-KdV system.Semi-implicit time discretization is adopted to relax the time step constraint from the high order spatial derivatives.Numerical results for accuracy tests of stationary traveling soliton,and the collision of solitons are shown.Numerical experiments illustrate the accuracy and capability of the method.

Vlasov-Fokker-Planck Simulations for High-Power Laser-Plasma Interactions

A review is presented on our recent Vlasov-Fokker-Planck(VFP)simulation code development and applications for high-power laser-plasma interactions.Numerical schemes are described for solving the kinetic VFP equation with both electronelectron and electron-ion collisions in one-spatial and two-velocity(1D2V)coordinates.They are based on the positive and flux conservation method and the finite volume method,and these twomethods can insure the particle number conservation.Our simulation code can deal with problems in high-power laser/beam-plasma interactions,where highly non-Maxwellian electron distribution functions usually develop and the widely-used perturbation theories with the weak anisotropy assumption of the electron distribution function are no longer in point.We present some new results on three typical problems:firstly the plasma current generation in strong direct current electric fields beyond Spitzer-H¨arm’s transport theory,secondly the inverse bremsstrahlung absorption at high laser intensity beyond Langdon’s theory,and thirdly the heat transport with steep temperature and/or density gradients in laser-produced plasma.Finally,numerical parameters,performance,the particle number conservation,and the energy conservation in these simulations are provided.

Particle-number conserving analysis for the 2-quasiparticle and high-K multi-quasiparticle states in doubly-odd ^(174,176)Lu

Two-quasiparticle bands and low-lying excited high-K four-, six-, and eight-quasiparticle bands in the doubly-odd 174, 176Lu are analyzed by using the cranked shell model （CSM） with the pairing correlations treated by a particle-number conserving （PNC） method, in which the blocking effects are taken into account exactly. The proton and neutron Nilsson level schemes for 174, 176Lu are taken from the adjacent odd-A Lu and Hf isotopes, which are adopted to reproduce the experimental bandhead energies of the one-quasiproton and one-quasineutron bands of these odd-A Lu and Hf nuclei, respectively. Once the quasiparticle configurations are determined, the experimental bandhead energies and the moments of inertia of these two- and multi-quasiparticle bands are well reproduced by PNC-CSM calculations. The Coriolis mixing of the low-K （K=｜Ω1-Ω2｜） two-quasiparticle band of the Gallagher-Moszkowski doublet with one nucleon in the Ω = 1/2 orbital is analyzed.

An implicit parallel UGKS solver for flows covering various regimes

This paper presents an engineering-oriented UGKS solver package developed in China Aerodynamics Research and Development Center(CARDC).The solver is programmed in Fortran language and uses structured body-fitted mesh,aiming for predicting aerodynamic and aerothermodynamics characteristics in flows covering various regimes on complex three-dimensional configurations.The conservative discrete ordinate method and implicit implementation are incorporated.Meanwhile,a local mesh refinement technique in the velocity space is developed.The parallel strategies include MPI and OpenMP.Test cases include a wedge,a cylinder,a 2D blunt cone,a sphere,and a X38-like vehicle.Good agreements with experimental or DSMC results have been achieved.

Particle-number conserving analysis of the high-K multi-quasiparticle bands in ^(179)Re

The experimentally observed ten rotational bands in 179Re are analyzed with the particle-number conserving method for treating the cranked shell model with pairing interaction, in which the blocking effects are taken into account exactly. The experimental moments of inertia of these bands are reproduced quite well by our calculations with no free parameter and the deformation driving effects are discussed. The bandhead energies and the variation in the occupation probability of each cranked orbital are also analyzed.

DynamicalMotion Driven by Periodic Forcing on an Open Elastic Tube in Fluid

We present a three dimensional model of an open elastic tube immersed in fluid to understand valveless pumping mechanism.A fluid-tube interaction problem is simulated by the volume conserved immersed boundarymethodwhich prevents the generation of spurious velocity field near the tube and local cluster of the tube surface.In order to explain pumping phenomena without valves,average net flow ismeasured by changing parameter values such as pumping frequency,compression duration,and pumping amplitude.Some frequencies that make the system reach maximal or minimal net flow are selected to study case by case.We also study the effectiveness of fluid mixing using the Shannon entropy increase rate.

A Gas-Kinetic Unified Algorithm for Non-Equilibrium Polyatomic Gas Flows Covering Various Flow Regimes

In this paper,a gas-kinetic unified algorithm(GKUA)is developed to investigate the non-equilibrium polyatomic gas flows covering various regimes.Based on the ellipsoidal statistical model with rotational energy excitation,the computable modelling equation is presented by unifying expressions on the molecular collision relaxing parameter and the local equilibrium distribution function.By constructing the corresponding conservative discrete velocity ordinate method for this model,the conservative properties during the collision procedure are preserved at the discrete level by the numerical method,decreasing the computational storage and time.Explicit and implicit lower-upper symmetric Gauss-Seidel schemes are constructed to solve the discrete hyperbolic conservation equations directly.Applying the new GKUA,some numerical examples are simulated,including the Sod Riemann problem,homogeneous flow rotational relaxation,normal shock structure,Fourier and Couette flows,supersonic flows past a circular cylinder,and hypersonic flow around a plate placed normally.The results obtained by the analytic,experimental,direct simulation Monte Carlo method,and other measurements in references are compared with the GKUA results,which are in good agreement,demonstrating the high accuracy of the present algorithm.Especially,some polyatomic gas non-equilibrium phenomena are observed and analysed by solving the Boltzmann-type velocity distribution function equation covering various flow regimes.