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.     

Multi-symplectic method for generalized Boussinesq equation

The generalized Boussinesq equation that represents a group of important nonlinear equations possesses many interesting properties. Multi-symplectic formulations of the generalized Boussinesq equation in the Hamilton space are introduced in this paper. And then an implicit multi-symplectic scheme equivalent to the multi-symplectic Box scheme is constructed to solve the partial differential equations （PDEs） derived from the generalized Boussinesq equation. Finally, the numerical experiments on the soliton solutions of the generalized Boussinesq equation are reported. The results show that the multi-symplectic method is an efficient algorithm with excellent long-time numerical behaviors for nonlinear partial differential equations.     

Multi-symplectic method for the generalized (2+1)-dimensional KdV-mKdV equation

In the present paper,a general solution involving three arbitrary functions for the generalized(2+1)dimensional KdV-mKdV equation,which is derived fromthe generalized(1+1)-dimensional KdV-mKdV equation,is first introduced by means of the Wiess,Tabor,Carnevale(WTC) truncation method.And then multisymplectic formulations with several conservation lawstaken into account are presented for the generalized(2+1)dimensional KdV-mKdV equation based on the multisymplectic theory of Bridges.Subsequently,in order tosimulate the periodic wave solutions in terms of rationalfunctions of the Jacobi elliptic functions derived from thegeneral solution,a semi-implicit multi-symplectic schemeis constructed that is equivalent to the Preissmann scheme.From the results of the numerical experiments,we can conclude that the multi-symplectic schemes can accurately simulate the periodic wave solutions of the generalized(2+1)dimensional KdV-mKdV equation while preserve approximately the conservation laws.     

Structure-preserving properties of Strmer-Verlet scheme for mathematical pendulum

The structure-preserving property, in both the time domain and the frequency domain, is an important index for evaluating validity of a numerical method. Even in the known structure-preserving methods such as the symplectic method, the inherent conservation law in the frequency domain is hardly conserved. By considering a mathematical pendulum model, a Strmer-Verlet scheme is first constructed in a Hamiltonian framework. The conservation law of the Strmer-Verlet scheme is derived, including the total energy expressed in the time domain and periodicity in the frequency domain. To track the structure-preserving properties of the Strmer-Verlet scheme associated with the conservation law, the motion of the mathematical pendulum is simulated with different time step lengths. The numerical results illustrate that the Strmer-Verlet scheme can preserve the total energy of the model but cannot preserve periodicity at all. A phase correction is performed for the Strmer-Verlet scheme. The results imply that the phase correction can improve the conservative property of periodicity of the Strmer-Verlet scheme.     

Gauge principle and variational formulation for flows of an ideal fluid

A gauge principle is applied to mass flows of an ideal compressible fluid subject to Galilei transformation. A free-field Lagrangian defined at the outset is invariant with respeet to global SO(3) gauge transformations as well as Galilei transformations. The action principle leads to the equation of potential flows under constraint of a continuity equation. However, the irrotational flow is not invariant with respect to local SO(3) gauge transformations. According to the gauge principle, a gauge-covariant derivative is defined by introducing a new gauge field. Galilei invariance of the derivative requires the gauge field to coincide with the vorticity, i.e. the curl of the velocity field. A full gauge-covariant variational formulation is proposed on the basis of the Hamilton‘‘s principle and an assoicated Lagrangian. By means of an isentropic material variation taking into account individual particle motion, the Euler‘‘s equation of motion is derived for isentropic flows by using the covariant derivative. Noether‘‘s law associated with global SO(3) gauge invariance leads to the conservation of total angular momentum. In addition, the Lagrangian has a local symmetry of particle permutation which results in local conservation law equivalent to the vorticity equation.     

SOLUTIONS FOR CYLINDRICAL CAVITY IN SATURATED THERMOPOROELASTIC MEDIUM

Based on the thermodynamics of irreversible processes, the mass conservation equation and heat energy balance equation are established. The governing equations of thermal consolidation for homogeneous isotropic materials are presented, accounting for the coupling effects of the temperature, stress and displacement fields. The case of a saturated medium with a long cylindrical cavity subjected to a variable thermal loading and a variable hydrostatic pressure （or a variable radial water flux） with time is considered. The analytical solutions are derived in the Laplace transform space. Then, the time domain solutions are obtained by a numerical inversion scheme. The results of a typical example indicate that thermodynamically coupled effects have considerable influences on thermal responses.     

Analysis of a two-grid method for semiconductor device problem

The mathematical model of a semiconductor device is governed by a system of quasi-linear partial differential equations.The electric potential equation is approximated by a mixed finite element method,and the concentration equations are approximated by a standard Galerkin method.We estimate the error of the numerical solutions in the sense of the Lqnorm.To linearize the full discrete scheme of the problem,we present an efficient two-grid method based on the idea of Newton iteration.The main procedures are to solve the small scaled nonlinear equations on the coarse grid and then deal with the linear equations on the fine grid.Error estimation for the two-grid solutions is analyzed in detail.It is shown that this method still achieves asymptotically optimal approximations as long as a mesh size satisfies H=O(h^1/2).Numerical experiments are given to illustrate the efficiency of the two-grid method.     

Conservation Law and Application of J-Integral in Multi-Materials

The conservation law of J-integral in two-media with a crack paralleling to the interface of the two media was firstly proved by analytical and numerical finite element method. Then a schedule model was established that an interface crack is inserted in four media. According to the J-integral conservation law on multi-media, the energy release ratio of Ⅰ-type crack was considered to be conservation when the middle medium layers are very thin. And the conservation law was also convinced by numerical method. By means of the dimension analysis on the model, the asymptotic results and formula calculating the energy release ratio and complex stress intensity factor are presented.     

Direct modeling for computational fluid dynamics

All fluid dynamic equations are valid under their modeling scales, such as the particle mean free path and mean collision time scale of the Boltzmann equation and the hydrodynamic scale of the Navier–Stokes(NS) equations.The current computational fluid dynamics(CFD) focuses on the numerical solution of partial differential equations(PDEs), and its aim is to get the accurate solution of these governing equations. Under such a CFD practice, it is hard to develop a unified scheme that covers flow physics from kinetic to hydrodynamic scales continuously because there is no such governing equation which could make a smooth transition from the Boltzmann to the NS modeling. The study of fluid dynamics needs to go beyond the traditional numerical partial differential equations. The emerging engineering applications, such as air-vehicle design for near-space flight and flow and heat transfer in micro-devices, do require further expansion of the concept of gas dynamics to a larger domain of physical reality, rather than the traditional distinguishable governing equations. At the current stage, the non-equilibrium flow physics has not yet been well explored or clearly understood due to the lack of appropriate tools.Unfortunately, under the current numerical PDE approach, it is hard to develop such a meaningful tool due to the absence of valid PDEs. In order to construct multiscale and multiphysics simulation methods similar to the modeling process of constructing the Boltzmann or the NS governing equations, the development of a numerical algorithm should be based on the first principle of physical modeling. In this paper, instead of following the traditional numerical PDE path, we introduce direct modeling as a principle for CFD algorithm development. Since all computations are conducted in a discretized space with limited cell resolution, the flow physics to be modeled has to be done in the mesh size and time step scales.Here, the CFD is more or less a direct construction of discrete numerical evolution equations, where the mesh size and time step will play dynamic roles in the modeling process.With the variation of the ratio between mesh size and local particle mean free path, the scheme will capture flow physics from the kinetic particle transport and collision to the hydrodynamic wave propagation. Based on the direct modeling, a continuous dynamics of flow motion will be captured in the unified gas-kinetic scheme. This scheme can be faithfully used to study the unexplored non-equilibrium flow physics in the transition regime.     

Numerical simulation of standing wave with 3D predictor-corrector finite difference method for potential flow equations

A three-dimensional （3D） predictor-corrector finite difference method for standing wave is developed. It is applied to solve the 3D nonlinear potential flow equa- tions with a free surface. The 3D irregular tank is mapped onto a fixed cubic tank through the proper coordinate transform schemes. The cubic tank is distributed by the staggered meshgrid, and the staggered meshgrid is used to denote the variables of the flow field. The predictor-corrector finite difference method is given to develop the difference equa- tions of the dynamic boundary equation and kinematic boundary equation. Experimental results show that, using the finite difference method of the predictor-corrector scheme, the numerical solutions agree well with the published results. The wave profiles of the standing wave with different amplitudes and wave lengths are studied. The numerical solutions are also analyzed and presented graphically.     

Analyzing dynamic response of non-homogeneous string fixed at both ends

To survey the local conservation properties of complex dynamic problems, a structure-preserving numerical method, named as generalized multi-symplectic method, is proposed to analyze the dynamic response of a non-homogeneous string fixed at both ends. Firstly, based on the multi-symplectic idea, a generalized multi-symplectic form derived from the vibration equation of a non-homogeneous string is presented. Secondly, several conservation laws are deduced from the generalized multi-symplectic form to illustrate the local properties of the system. Thirdly, a centered box difference scheme satisfying the discrete local momentum conservation law exactly, named as a generalized multi-symplectic scheme, is constructed to analyze the dynamic response of the non-homogeneous string fixed at both ends. Finally, numerical experiments on the generalized multi-symplectic scheme are reported. The results illustrate the high accuracy, the good local conservation properties as well as the excellent long-time numerical behavior of the generalized multi-symplectic scheme well.     

THE MULTI-SYMPLECTIC ALGORITHM FOR “GOOD” BOUSSINESQ EQUATION

The multi-symplectic formulations of the “Good” Boussinesq equation were considered. For the multi-symplectic formulation, a new fifteen-point difference scheme which is equivalent to the multi-symplectic Preissman integrator was derived. The numerical experiments show that the multi-symplectic scheme have excellent long-time numerical behavior. Foundation items: the Foundation for Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences; the Natural Science Foundation of Huaqiao University. Biography: ZENG Wen-ping (1940-), Professor (E-mail: qmz@1sec.cc.ac.cn)     

New explicit multi-symplectic scheme for nonlinear wave equation简

Based on splitting multi-symplectic structures, a new multi-symplectic scheme is proposed and applied to a nonlinear wave equation. The explicit multi-symplectic scheme of the nonlinear wave equation is obtained, and the corresponding multi-symplectic conservation property is proved. The backward error analysis shows that the explicit multi-symplectic scheme has good accuracy. The sine-Gordon equation and the Klein-Gordon equation are simulated by an explicit multi-symplectic scheme. The numerical results show that the new explicit multi-symplectic scheme can well simulate the solitary wave behaviors of the nonlinear wave equation and approximately preserve the relative energy error of the equation.     

DIFFERENCE SCHEME FOR TWO-PHASE FLOW

A numerical method for two-phase flow with hydrodynamics behavior was considered. The nonconservative hyperbolic governing equations proposed by Saurel and Gallout were adopted. Dissipative effects were neglected but they could be included in the model without major difficulties. Based on the opinion proposed by Abgrall that “a two phase system, uniform in velocity and pressure at t = 0 will be uniform on the same variable during its temporal evolution“, a simple accurate and fully Eulerian numerical method was presented for the simulation of multiphase compressible flows in hydrodynamic regime. The numerical method relies on Godunov-typescheme, with HLLC and Lax-Friedrichs type approximate Riemann solvers for the resolution of conservation equations, and nonconservative equation. Speed relaxation and pressure relaxation processes were introduced to account for the interaction between the phases. Test problem was presented in one space dimension which illustrated that our scheme is accurate, stable and oscillation free.     

MIXED FINITE ELEMENT METHODS FOR THE SHALLOWWATER EQUATIONS INCLUDING CURRENT AND SILTSEDIMENTATION （Ⅱ）——THE DISCRETE-TIMECASE ALONG CHARACTERISTICS

The mixed finite element(MFE) methods for a shallow water equation systemconsisting of water dynamics equations, silt transport equation, and the equation of bottomtopography change were derived. A fully discrete MFE scheme for the discrete-time alongcharacteristics is presented and error estimates are established. The existence andconvergence of MFE solution of the discrete current velocity, elevation of the bottomtopography, thickness of fluid column, and mass rate of sediment is demonstrated.     

非线性弹性杆中纵波传播过程的数值模拟

基于Hamilton空间体系的多辛理论研究了包含材料非线性效应和几何弥散效应的非线性弹性杆中纵波传播问题。导出了其Bridges意义下的多辛形式及其多种守恒律,并构造了等价于Box多辛格式的隐式多辛格式对不考虑材料非线性效应和几何弥散效应、只考虑材料非线性效应、只考虑几何弥散效应、同时考虑材料非线性效应和几何弥散效应四种情况下不同截面参数的圆杆中的纵波传播过程进行数值模拟,模拟结果不仅全面地反映了非线性效应和几何弥散效应对纵波传播的影响,而且也反映了多辛方法的两大优点:精确的保持多种守恒律和良好的长时间数值行为。