Semi-Implicit Scheme to Solve Allen-Cahn Equation with Different Boundary Conditions

The aim of this paper is to give an appropriate numerical method to solve Allen-Cahn equation, with Dirichlet or Neumann boundary condition. The time discretization involves an explicit scheme for the nonlinear part of the operator and an implicit Euler discretization of the linear part. Finite difference schemes are used for the spatial part. This finally leads to the numerical solution of a sparse linear system that can be solved efficiently.

A three-dimensional semi-implicit unstructured grid finite volume ocean model

A new three-dimensional semi-implicit finite-volume ocean model has been developed for simulating the coastal ocean circulation, which is based on the staggered C-unstructured non-orthogonal grid in the hor- izontal direction and z-level grid in the vertical direction. The three-dimensional model is discretized by the semi-implicit finite-volume method, in that the free-surface and the vertical diffusion are semi-implicit, thereby removing stability limitations associated with the surface gravity wave and vertical diffusion terms. The remaining terms in the momentum equations are discretized explicitly by an integral method. The partial cell method is used for resolving topography, which enables the model to better represent irregular topography. The model has been tested against analytical cases for wind and tidal oscillation circulation, and is applied to simulating the tidal flow in the Bohal Sea. The results are in good agreement both with the analytical solutions and measurement results.

Moving-Particle Semi-Implicit Method for Vortex Patterns and Roll Damping of 2D Ship Sections

A meshless method, Moving-Particle Semi-hnplicit Method （MPS） is presented in this paper to simulate the rolling of different 2D ship sections. Sections S. S. 0.5, S.S. 5.0 and S. S. 7.0 of series 60 with CB = 0.6 are chosen for the simulation. It shows that the result of MPS is very close to results of experiments or mesh-numerical simulations. In the simulation of MPS, vortices are found periodically in bilges of ship sections. In section S. S. 5.0 and section S. S. 7.0, which are close to the middle ship, two little vortices are found at different bilges of the section, in section S. S. 0.5, which is close to the bow, only one big vortex is found at the bottom of the section, these vortices patterns are consistent with the theory of Ikeda. The distribution of shear stress and pressure on the rolling hull of ship section is calculated. When vortices are in bilges of the section, the sign clmnge of pressure can be found, but in section S. S. 0.5, there is no sign change of pressure because only one vortex in the bottom of the section. With shear stress distribution, it can be found the shear stress in bilges is bigger than that at other part of the ship section. As the free surface is considered, the shear stress of both sides near the free surface is close to zero and even sign changed.

Numerical simulation of viscous liquid sloshing by moving-particle semi-implicit method

A meshless numerical simulation method, the moving-particle semi-implicit method （MPS） is presented in this paper to study the sloshing phenomenon in ocean and naval engineering. As a meshless method, MPS uses particles to replace the mesh in traditional methods, the governing equations are discretized by virtue of the relationship of particles, and the Poisson equation of pressure is solved by incomplete Cholesky conjugate gradient method （ICCG）, the free surface is tracked by the change of numerical density. A numerical experiment of viscous liquid sloshing tank was presented and compared with the result got by the difference method with the VOF, and an additional modification step was added to make the simulation more stable. The results show that the MPS method is suitable for the simulation of viscous liquid sloshing, with the advantage in arranging the particles easily, especially on some complex curved surface.

An improved semi-implicit direct kinetics method for transient analysis of nuclear reactors

Semi-implicit direct kinetics(SIDK)is an innovative method for the temporal discretization of neutronic equations proposed by J.Banfield.The key approximation of the SIDK method is to substitute a timeaveraged quantity for the fission source term in the delayed neutron differential equations.Hence,these equations are decoupled from prompt neutron equations and an explicit analytical representation of precursor groups is obtained,which leads to a significant reduction in computational cost.As the fission source is not known in a time step,the original study suggested using a constant quantity pertaining to the previous time step for this purpose,and a reduction in the size of the time step was proposed to lessen the imposed errors.However,this remedy notably diminishes the main advantage of the SIDK method.We discerned that if the original method is properly introduced into the algorithm of the point-implicit solver along with some modifications,the mentioned drawbacks will be mitigated adequately.To test this idea,a novel multigroup,multi-dimensional diffusion code using the finitevolume method and a point-implicit solver is developed which works in both transient and steady states.In addition to the SIDK,two other kinetic methods,i.e.,direct kinetics and higher-order backward discretization,are programmed into the diffusion code for comparison with the proposed model.The final code is tested at different conditions of two well-known transient benchmark problems.Results indicate that while the accuracy of the improved SIDK is closely comparable with the best available kinetic methods,it reduces the total time required for computation by up to 24%.

Large deformation simulations of geomaterials using moving particle semi-implicit method

Numerical simulation tools are required to describe large deformations of geomaterials for evaluating the risk of geo-disasters. This study focused on moving particle semi-implicit(MPS) method, which is a Lagrangian gridless particle method, and investigated its performance and stability to simulate large deformation of geomaterials. A calculation method was developed using geomaterials modeled as Bingham fluids to improve the original MPS method and enhance its stability. Two numerical tests showed that results from the improved MPS method was in good agreement with the theoretical value.Furthermore, numerical simulations were calibrated by laboratory experiments. It showed that the simulation results matched well with the experimentally observed free-surface configurations for flowing sand. In addition, the model could generally predict the time-history of the impact force. The MPS method could be a useful tool to evaluate large deformation of geomaterials.

Moving particle semi-implicit simulation on the molten Wood's metal downward relocation process

In the case of a severe accident involving nuclear reactors,an important aspect that should be considered is the leakage of molten material from the inside of the reactor into the environment.These molten materials damage other reactor components,such as electrical tubes,grid plates and core catchers.In this study,the moving particle semi-implicit(MPS)method is adopted and improved to analyze the twodimensional downward relocation process of molten Wood’s metal as a representation of molten material in a nuclear reactor.The molten material impinges the Wood’s metal plate(WMP),which is mounted on a rigid dummy stainless steel in a cylindrical test vessel.The breaching process occurs because of heat transfer between the molten material and WMP.The formed breach areas were in good agreement with the experimental results,and they showed that the molten Wood’s metal spread above the WMP.The solid WMP fraction decreased with time until it reached the termination time of the simulation.The present results show that the MPS method can be applied to simulate and analyze the downward relocation process of molten material in the grid plate of a nuclear reactor.

The Semi-implicit Euler Method for Stochastic Pantograph Equations with Jumps

In this paper,we present the semi-implicit Euler（SIE）numerical solution for stochastic pantograph equations with jumps and prove that the SIE approximation solution converges to the exact solution in the mean-square sense under the Local Lipschitz condition.

Stability of Semi-implicit Finite Volume Scheme for Level Set Like Equation

We study numerical methods for level set like equations arising in image processing and curve evolution problems. Semi-implicit finite volume-element type schemes are constructed for the general level set like equation （image selective smoothing model） given by Alvarez et al. （Alvarez L, Lions P L, Morel J M. Image selective smoothing and edge detection by nonlinear diffusion II. SIAM J. Numer. Anal., 1992, 29： 845-866）. Through the reasonable semi-implicit discretization in time and co-volume method for space approximation, we give finite volume schemes, unconditionally stable in L∞ and W1＇2 （W1＇1） sense in isotropic （anisotropic） diffu- sion domain.

A Semi-Implicit Scheme of Lattice Boltzmann Method for Two Dimensional Cavity Flow Simulation

The calculation sequence of collision, propagation and macroscopic variables is not very clear in lattice Boltzmann method (LBM) code implementation. According to the definition, three steps should be computed on all nodes respectively, which mean three loops are needed. While the “pull” scheme makes the only one loop possible for coding, this is called semi-implicit scheme in this study. The accuracy and efficiency of semi-implicit scheme are discussed in detail through the simulation of lid-driven cavity flow. Non-equilibrium extrapolation scheme is adopted on the boundary of simulation area. The results are compared with two classic articles, which show that semi-implicit scheme has good agreement with the classic scheme. When Re is less than 3000, the iterations steps of semi-scheme can be decreased by about 30% though comparing the semi-implicit scheme with standard scheme containing three loops. As the Re increases into more than 3400, the standard scheme is not converged. On the contrary, the iterations of semi-implicit scheme are approximately linear to Re.

High Order Semi-implicit Multistep Methods for Time-Dependent Partial Differential Equations

We consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form,typically used in implicit-explicit(IMEX)methods,is not possible.As shown in Boscarino et al.(J.Sci.Comput.68:975-1001,2016)for Runge-Kutta methods,these semi-implicit techniques give a great flexibility,and allow,in many cases,the construction of simple linearly implicit schemes with no need of iterative solvers.In this work,we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype lineal'advection-diffusion equation and in the setting of strong stability preserving(SSP)methods.Our findings are demonstrated on several examples,including nonlinear reaction-diffusion and convection-diffusion problems.

A Novel Staggered Semi-implicit Space-Time Discontinuous Galerkin Method for the Incompressible Navier-Stokes Equations

A new high-order accurate staggered semi-implicit space-time discontinuous Galerkin(DG)method is presented for the simulation of viscous incompressible flows on unstructured triangular grids in two space dimensions.The staggered DG scheme defines the discrete pressure on the primal triangular mesh,while the discrete velocity is defined on a staggered edge-based dual quadrilateral mesh.In this paper,a new pair of equal-order-interpolation velocity-pressure finite elements is proposed.On the primary triangular mesh(the pressure elements),the basis functions are piecewise polynomials of degree N and are allowed to jump on the boundaries of each triangle.On the dual mesh instead(the velocity elements),the basis functions consist in the union of piecewise polynomials of degree N on the two subtriangles that compose each quadrilateral and are allowed to jump only on the dual element boundaries,while they are continuous inside.In other words,the basis functions on the dual mesh arc built by continuous finite elements on the subtriangles.This choice allows the construction of an efficient,quadrature-free and memory saving algorithm.In our coupled space-time pressure correction formulation for the incompressible Navier-Stokes equations,the arbitrary high order of accuracy in time is achieved through tire use of time-dependent test and basis functions,in combination with simple and efficient Picard iterations.Several numerical tests on classical benchmarks confirm that the proposed method outperforms existing staggered semi-implicit space-time DG schemes,not only from a computer memory point of view,but also concerning the computational time.

A Second-Order Semi-Implicit Method for the Inertial Landau-Lifshitz-Gilbert Equation

Electron spins in magnetic materials have preferred orientations collectively and generate the macroscopic magnetization.Its dynamics spans over a wide range of timescales from femtosecond to picosecond,and then to nanosecond.The Landau-Lifshitz-Gilbert(LLG)equation has been widely used in micromagnetics simulations over decades.Recent theoretical and experimental advances have shown that the inertia of magnetization emerges at sub-picosecond timescales and contributes significantly to the ultrafast magnetization dynamics,which cannot be captured intrinsically by the LLG equation.Therefore,as a generalization,the inertial LLG(iLLG)equation is proposed to model the ultrafast magnetization dynamics.Mathematically,the LLG equation is a nonlinear system of parabolic type with(possible)degeneracy.However,the iLLG equation is a nonlinear system of mixed hyperbolic-parabolic type with degeneracy,and exhibits more complicated structures.It behaves as a hyperbolic system at sub-picosecond timescales,while behaves as a parabolic system at larger timescales spanning from picosecond to nanosecond.Such hybrid behaviors impose additional difficulties on designing efficient numerical methods for the iLLG equation.In this work,we propose a second-order semiimplicit scheme to solve the iLLG equation.The second-order temporal derivative of magnetization is approximated by the standard centered difference scheme,and the first-order temporal derivative is approximated by the midpoint scheme involving three time steps.The nonlinear terms are treated semi-implicitly using one-sided interpolation with second-order accuracy.At each time step,the unconditionally unique solvability of the unsymmetric linear system is proved with detailed discussions on the condition number.Numerically,the second-order accuracy of the proposed method in both time and space is verified.At sub-picosecond timescales,the inertial effect of ferromagnetics is observed in micromagnetics simulations,in consistency with the hyperbolic property of the iLLG model;at nanosecond timescales,the results of the iLLG model are in nice agreements with those of the LLG model,in consistency with the parabolic feature of the iLLG model.

考虑小应变刚度特性的软土边界面模型与应用

基坑开挖导致周边土体经历复杂的加卸载应力路径,传统的本构模型难以同时反映此过程中土体的超固结和小应变刚度特性。在现有的边界面模型中,引入土体小应变刚度特性,提出软土边界面模型小应变刚度修正方法。基于一种高效的半隐式应力更新算法,将改进后的边界面模型采用UMAT子程序二次开发嵌入有限元软件ABAQUS中,并应用于某地铁车站深基坑开挖工程的数值模拟。分析结果表明,采用改进模型准确地反映了上海地区典型土层的小应变状态本构关系和土体的应变刚度相关性;由于改进的边界面模型能够同时反映土体超固结和小应变刚度特性,计算的围护结构变形与现场监测数据吻合较好。

油液环境中有载分接开关快速机构切换特性

利用基于移动粒子半隐式方法的Particleworks,分析有载分接开关快速机构在油液环境中的切换特性.介绍有载分接开关快速机构的工作原理,设计有载分接开关快速机构的机械结构,给出快速机构在油液环境下的动力学计算基础.建立该有载分接开关快速机构在油液环境中的动力学仿真模型,对照了该快速机构在有无变压器油液环境中的切换特性差异,得到驱动电机的驱动扭矩要求大于40 N·m.分析油液黏度和储能弹簧刚度对该快速机构切换特性的影响,得到快速机构在油液环境下切换过程中飞轮受到的油液阻力矩的变化规律,通过流场分析揭示了飞轮受到的阻力矩变化和波动的原因.

半隐式延迟解耦的开关动作高效处理方法

传统电磁暂态仿真采用详细模型建模电压源型换流器(voltage source converter,VSC)并采用双插值法(double interpolation method,DIM)处理步长内的开关事件,由于详细模型的导纳阵时变,DIM算法流程复杂,在进行大电网仿真时严重影响计算效率。针对这一问题,应用半隐式延时解耦原理建模VSC,并对步长内开关事件的处理方法进行研究,通过对开关函数在步长内的占空比进行等值代替DIM,提高仿真效率。首先,从VSC状态方程出发,得到VSC解耦模型,实现换流器的并行计算,提高计算效率;其次,分析了不处理开关事件时累计误差的产生机理,并给出了计算时序和开关函数等值占空比的计算方法,保证仿真精度;最后,以PSCAD的仿真波形为基准,验证了所提方法的精度和有效性,讨论了所提方法的方法特点。

A semi-implicit semi-Lagrangian global nonhydrostatic model and the polar discretization scheme

The Global/Regional Assimilation and PrEdiction System(GRAPES) is a newly developed global non-hydrostatic numerical prediction model,which will become the next generation medium-range opera-tional model at China Meteorological Administration(CMA).The dynamic framework of GRAPES is featuring with fully compressible equations,nonhydrostatic or hydrostatic optionally,two-level time semi-Lagrangian and semi-implicit time integration,Charney-Phillips vertical staggering,and complex three-dimensional pre-conditioned Helmholtz solver,etc.Concerning the singularity of horizontal momentum equations at the poles,the polar discretization schemes are described,which include adoption of Arakawa C horizontal grid with ν at poles,incorporation of polar filtering to maintain the computational stability,the correction to Helmholtz equation near the poles,as well as the treatment of semi-Lagrangian interpolation to improve the departure point accuracy,etc.The balanced flow tests validate the rationality of the treatment of semi-Lagrangian departure point calculation and the polar discretization during long time integration.Held and Suarez tests show that the conservation proper-ties of GRAPES model are quite good.

A SEMI-IMPLICIT 3-D NUMERICAL MODEL USING SIGAM-COORDINATE FOR NON-HYDROSTATIC PRESSURE FREE-SURFACE FLOWS

A 3-D numerical formulation is proposed on the horizontal Cartesian, vertical sigma-coordinate grid for modeling non-hydrostatic pressure flee-surface flows. The pressure decomposition technique and 0 semi-implicit method are used, with the solution procedure being split into two steps. First, with the implicit parts of non-hydrostatic pressures excluded, the provisional velocity field and free surface are obtained by solving a 2-D Poisson equation. Second, the theory of the differential operator is employed to derive the 3-D Poisson equation for non-hydrostatic pressures, which is solved to obtain the non-hydrostatic pressures and to update the provisional velocity field. When the non-orthogonal sigma-coordinate transformation is introduced, additional terms come into being, resulting in a 15-diagonal, diagonally dominant but unsymmetric linear system in the 3-D Poisson equation for non-hydrostatic pressures. The Biconjugate Gradient Stabilized （BiCGstab） method is used to solve the resulting 3-D unsymmetric linear system instead of the conjugate gradient method, which can only be used for symmetric, positive-definite linear systems. Three test cases are used for validations. The successful simulations of the small-amplitude wave, a supercritical flow over a ramp and a turbulent flow in the open channel indicate that the new model can simulate well non-hydrostatic flows, supercritical flows and turbulent flows.

THE FORMULATION OF A PERFECT SQUARE CONSERVATIVE SEMI-IMPLICIT TIME DIFFERENCE SCHEME AND ITS PRELIMINARY CHECK

Quantitative studies of scientific problems require solving correspondent mathematical models. Although a great deal of mathematical models of evolutional problems are set up under continuous space-time meaning, they usually had to be solved numerically after space-time discretization because nonlinear mathematical models except some

Semi-Implicit Spectral Deferred Correction Method Based on the Invariant Energy Quadratization Approach for Phase Field Problems

This paper presents a high order time discretization method by combining the semi-implicit spectral deferred correction method with energy stable linear schemes to simulate a series of phase field problems.We start with the linear scheme,which is based on the invariant energy quadratization approach and is proved to be linear unconditionally energy stable.The scheme also takes advantage of avoiding nonlinear iteration and the restriction of time step to guarantee the nonlinear system uniquely solvable.Moreover,the scheme leads to linear algebraic system to solve at each iteration,and we employ the multigrid solver to solve it efficiently.Numerical re-sults are given to illustrate that the combination of local discontinuous Galerkin(LDG)spatial discretization and the high order temporal scheme is a practical,accurate and efficient simulation tool when solving phase field problems.Namely,we can obtain high order accuracy in both time and space by solving some simple linear algebraic equations.