Some Splitting Methods for Equations of Geophysical Fluid Dynamics

In this paper, equations of atmospheric and oceanic dynamics are reduced to a kind of evolutionary equation in operator form, based on which a conclusion that the separability of motion stages is relative is made and an issue that the tractional splitting methods established on the physical separability of the fast stage and the slow stage neglect the interaction between the two stages to some extent is shown. Also, three splitting patterns are summed up from the splitting methods in common use so that a comparison between them is carried out. The comparison shows that only the improved splitting pattern (ISP) can be in second order and keep the interaction well. Finally, the applications of some splitting methods on numerical simulations of typhoon tracks made clear that ISP owns the best effect and can save more than 80% CPU time.

Efficient Splitting Methods Based on Modified Potentials:Numerical Integration of Linear Parabolic Problems and Imaginary Time Propagation of the Schrodinger Equation

We present a new family of fourth-order splitting methods with positive coefficients especially tailored for the time integration of linear parabolic problems and,in particular,for the time dependent Schrodinger equation,both in real and imaginary time.They are based on the use of a double commutator and a modified processor,and are more efficient than other widely used schemes found in the literature.Moreover,for certain potentials,they achieve order six.Several examples in one,two and three dimensions clearly illustrate the computational advantages of the new schemes.

POSITIVE DEFINITE AND SEMI-DEFINITE SPLITTING METHODS FOR NON-HERMITIAN POSITIVE DEFINITE LINEAR SYSTEMS

In this paper, we further generalize the technique for constructing the normal （or pos- itive definite） and skew-Hermitian splitting iteration method for solving large sparse non- Hermitian positive definite system of linear equations. By introducing a new splitting, we establish a class of efficient iteration methods, called positive definite and semi-definite splitting （PPS） methods, and prove that the sequence produced by the PPS method con- verges unconditionally to the unique solution of the system. Moreover, we propose two kinds of typical practical choices of the PPS method and study the upper bound of the spectral radius of the iteration matrix. In addition, we show the optimal parameters such that the spectral radius achieves the minimum under certain conditions. Finally, some numerical examples are given to demonstrate the effectiveness of the considered methods.

Explicit Multi-Symplectic Splitting Methods for the Nonlinear Dirac Equation

In this paper,we propose two new explicit multi-symplectic splitting methods for the nonlinear Dirac(NLD)equation.Based on its multi-symplectic formulation,the NLD equation is split into one linear multi-symplectic system and one nonlinear infinite Hamiltonian system.Then multi-symplectic Fourier pseudospectral method and multi-symplectic Preissmann scheme are employed to discretize the linear subproblem,respectively.And the nonlinear subsystem is solved by a symplectic scheme.Finally,a composition method is applied to obtain the final schemes for the NLD equation.We find that the two proposed schemes preserve the total symplecticity and can be solved explicitly.Numerical experiments are presented to show the effectiveness of the proposed methods.

High-Order Decoupled and Bound Preserving Local Discontinuous Galerkin Methods for a Class of Chemotaxis Models

In this paper,we explore bound preserving and high-order accurate local discontinuous Galerkin(LDG)schemes to solve a class of chemotaxis models,including the classical Keller-Segel(KS)model and two other density-dependent problems.We use the convex splitting method,the variant energy quadratization method,and the scalar auxiliary variable method coupled with the LDG method to construct first-order temporal accurate schemes based on the gradient flow structure of the models.These semi-implicit schemes are decoupled,energy stable,and can be extended to high accuracy schemes using the semi-implicit spectral deferred correction method.Many bound preserving DG discretizations are only worked on explicit time integration methods and are difficult to get high-order accuracy.To overcome these difficulties,we use the Lagrange multipliers to enforce the implicit or semi-implicit LDG schemes to satisfy the bound constraints at each time step.This bound preserving limiter results in the Karush-Kuhn-Tucker condition,which can be solved by an efficient active set semi-smooth Newton method.Various numerical experiments illustrate the high-order accuracy and the effect of bound preserving.

Average vector field methods for the coupled Schrdinger KdV equations

The energy preserving average vector field （AVF） method is applied to the coupled Schr6dinger-KdV equations. Two energy preserving schemes are constructed by using Fourier pseudospectral method in space direction discretization. In order to accelerate our simulation, the split-step technique is used. The numerical experiments show that the non-splitting scheme and splitting scheme are both effective, and have excellent long time numerical behavior. The comparisons show that the splitting scheme is faster than the non-splitting scheme, but it is not as good as the non-splitting scheme in preserving the invariants.

Multi-Symplectic Splitting Method for Two-Dimensional Nonlinear Schrodinger Equation

Using the idea of splitting numerical methods and the multi-symplectic methods, we propose a multisymplectic splitting （MSS） method to solve the two-dimensional nonlinear Schrodinger equation （2D-NLSE） in this paper. It is further shown that the method constructed in this way preserve the global symplectieity exactly. Numerical experiments for the plane wave solution and singular solution of the 2D-NLSE show the accuracy and effectiveness of the proposed method.

An Improved Splitting Method

In this paper, an improved splitting method, based on the completely square-conservative explicit difference schemes, is established. Not only can the time-direction precision of this method be higher than that of the traditional splitting methods but also can the physical feature of mutual dependence of the fast and the slow stages that are calculated separately and splittingly be kept as well. Moreover, the method owns an universality, it can be generalized to other square-conservative difference schemes, such as the implicit and complete ones and the explicit and instantaneous ones. Good time benefits can be acquired when it is applied in the numerical simulations of the monthly mean currents of the South China Sea.

Computational Methods for Three Coupled Nonlinear Schrödinger Equations

In this work, we will derive numerical schemes for solving 3-coupled nonlinear Schrödinger equations using finite difference method and time splitting method combined with finite difference method. The resulting schemes are highly accurate, unconditionally stable. We use the exact single soliton solution and the conserved quantities to check the accuracy and the efficiency of the proposed schemes. Also, we use these methods to study the interaction dynamics of two solitons. It is found that both elastic and inelastic collision can take place under suitable parametric conditions. We have noticed that the inelastic collision of single solitons occurs in two different manners: enhancement or suppression of the amplitude.

THREE HIGH-ORDER SPLITTING SCHEMES FOR 3D TRANSPORT EQUATION

Two high-order splitting schemes based on the idea of the operators splitting method are given. The three-dimensional advection-diffusion equation was split into several one-dimensional equations that were solved by these two schemes, only three computational grid points were needed in each direction but the accuracy reaches the spatial fourth-order. The third scheme proposed is based on the classical ADI scheme and the accuracy of the advection term of it can reach the spatial fourth-order. Finally, two typical numerical experiments show that the solutions of these three schemes compare well with that given by the analytical solution when the Peclet number is not bigger than 5.

A Dimension-Splitting Variational Multiscale Element-Free Galerkin Method for Three-Dimensional Singularly Perturbed Convection-Diffusion Problems

By introducing the dimensional splitting(DS)method into the multiscale interpolating element-free Galerkin(VMIEFG)method,a dimension-splitting multiscale interpolating element-free Galerkin(DS-VMIEFG)method is proposed for three-dimensional(3D)singular perturbed convection-diffusion(SPCD)problems.In the DSVMIEFG method,the 3D problem is decomposed into a series of 2D problems by the DS method,and the discrete equations on the 2D splitting surface are obtained by the VMIEFG method.The improved interpolation-type moving least squares(IIMLS)method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems.The solved numerical example verifies the effectiveness of the method in this paper for the 3D SPCD problems.The numerical solution will gradually converge to the analytical solution with the increase in the number of nodes.For extremely small singular diffusion coefficients,the numerical solution will avoid numerical oscillation and has high computational stability.

A SUPERLINEARLY CONVERGENT SPLITTING FEASIBLE SEQUENTIAL QUADRATIC OPTIMIZATION METHOD FOR TWO-BLOCK LARGE-SCALE SMOOTH OPTIMIZATION

This paper discusses the two-block large-scale nonconvex optimization problem with general linear constraints.Based on the ideas of splitting and sequential quadratic optimization(SQO),a new feasible descent method for the discussed problem is proposed.First,we consider the problem of quadratic optimal(QO)approximation associated with the current feasible iteration point,and we split the QO into two small-scale QOs which can be solved in parallel.Second,a feasible descent direction for the problem is obtained and a new SQO-type method is proposed,namely,splitting feasible SQO(SF-SQO)method.Moreover,under suitable conditions,we analyse the global convergence,strong convergence and rate of superlinear convergence of the SF-SQO method.Finally,preliminary numerical experiments regarding the economic dispatch of a power system are carried out,and these show that the SF-SQO method is promising.

Explicit K-symplectic methods for nonseparable non-canonical Hamiltonian systems

We propose efficient numerical methods for nonseparable non-canonical Hamiltonian systems which are explicit,K-symplectic in the extended phase space with long time energy conservation properties. They are based on extending the original phase space to several copies of the phase space and imposing a mechanical restraint on the copies of the phase space. Explicit K-symplectic methods are constructed for two non-canonical Hamiltonian systems. Numerical tests show that the proposed methods exhibit good numerical performance in preserving the phase orbit and the energy of the system over long time, whereas higher order Runge–Kutta methods do not preserve these properties. Numerical tests also show that the K-symplectic methods exhibit better efficiency than that of the same order implicit symplectic, explicit and implicit symplectic methods for the original nonseparable non-canonical systems. On the other hand, the fourth order K-symplectic method is more efficient than the fourth order Yoshida’s method, the optimized partitioned Runge–Kutta and Runge–Kutta–Nystr ¨om explicit K-symplectic methods for the extended phase space Hamiltonians, but less efficient than the the optimized partitioned Runge–Kutta and Runge–Kutta–Nystr ¨om extended phase space symplectic-like methods with the midpoint permutation.

Multidimensional Numerical Simulation of Glow Discharge by Using the N-BEE-Time Splitting Method

In this work, a new numerical technique is proposed for the resolution of a fluid model based on three Boltzmann moments. The main purpose of this technique is to calculate electric and physical properties in the non-equilibrium electric discharge at low pressure. The transport and Poisson＇s equations form a self-consistent model. This equation system is written in cylindrical coordinates following the geometric shape of a plasma reactor. Our transport equation system is discretized using the finite volume approach and resolved by the N-BEE explicit scheme coupled to the time splitting method. This programming structure reduces computation time considerably. The 2D code is carried out and tested by comparing our results with those found in literature.

Multi-symplectic wavelet splitting method for the strongly coupled Schrodinger system

We propose a multi-symplectic wavelet splitting equations. Based on its mu]ti-symplectic formulation, method to solve the strongly coupled nonlinear SchrSdinger the strongly coupled nonlinear SchrSdinger equations can be split into one linear multi-symplectic subsystem and one nonlinear infinite-dimensional Hamiltonian subsystem. For the linear subsystem, the multi-symplectic wavelet collocation method and the symplectic Euler method are employed in spatial and temporal discretization, respectively. For the nonlinear subsystem, the mid-point symplectic scheme is used. Numerical simulations show the effectiveness of the proposed method during long-time numerical calculation.

SPLITTING MODULUS FINITE ELEMENT METHOD FOR ORTHOGONAL ANISOTROPIC PLATE BENGING

Splitting modulus variational principle in linear theory of solid mechanics was introduced, the principle for thin plate was derived, and splitting modulus finite element method of thin plate was established too. The distinctive feature of the splitting model is that its functional contains one or more arbitrary additional parameters, called splitting factors, so stiffness of the model can be adjusted by properly selecting the splitting factors. Examples show that splitting modulus method has high precision and the ability to conquer some ill-conditioned problems in usual finite elements. The cause why the new method could transform the ill-conditioned problems into well-conditioned problem, is analyzed finally.

Multichannel Blind CT Image Restoration via Variable Splitting and Alternating Direction Method

Computed tomography(CT) blurring caused by point spread function leads to errors in quantification and visualization. In this paper, multichannel blind CT image restoration is proposed to overcome the effect of point spread function. The main advantage from multichannel blind CT image restoration is to exploit the diversity and redundancy of information in different acquisitions. The proposed approach is based on a variable splitting to obtain an equivalent constrained optimization formulation, which is addressed with the alternating direction method of multipliers and simply implemented in the Fourier domain. Numerical experiments illustrate that our method obtains a higher average gain value of at least 1.21 d B in terms of Q metric than the other methods, and it requires only 7 iterations of alternating minimization to obtain a fast convergence.

The Operator Splitting Method for Black-Scholes Equation

The Operator Splitting method is applied to differential equations occurring as mathematical models in financial models. This paper provides various operator splitting methods to obtain an effective and accurate solution to the Black-Scholes equation with appropriate boundary conditions for a European option pricing problem. Finally brief comparisons of option prices are given by different models.

UPWIND SCHEME FOR IDEAL 2-D MHD FLOWS BASED ON UNSTRUCTURED MESH

An upwind scheme based on the unstructured mesh is developed to solve ideal 2-D magnetohydrodynamics （MHD） equations. The inviscid fluxes are approximated by using the modified advection upstream splitting method （AUSM） scheme, and a 5-stage explicit Runge-Kutta scheme is adopted in the time integration. To avoid the influence of the magnetic field divergence created during the simulation, the hyperbolic divergence cleaning method is introduced. The shock-capturing properties of the method are verified by solving the MHD shock-tube problem. Then the 2-D nozzle flow with the magnetic field is numerically simulated on the unstructured mesh. Computational results demonstrate the effects of the magnetic field and agree well with those from references.

NUMERICAL SIMULATIONS OF EVOLUTIONS FROM WALL PULSE TO TURBULENT COHE RENT STRUCTURES

The high order compact d if ference method is developed for solving the perturbation equations based on Navi er Stokes equations, and is used in studying complex evolution processes from w all negative pulse to the turbulent coherent structure in the channel flow. Th is method contains three dimensional coupling difference scheme with high accur acy and high resolution, and the high order time splitting methods. Compared with the general spectral method, the method can be used to research turbule nt coherent structure under more general boundary conditions and in flow domains . In this paper, the generation and evolution of the turbulent coherent structur es ind uced by wall pulse in the channel flow are simulated, and the basic characterist ics and rules of the turbulent coherent structure are shown. Computational r esults indicate that a wall negative pulse is more convenient than the resonant three wave model.