Alternating segment explicit-implicit scheme for nonlinear third-order KdV equation

A group of asymmetric difference schemes to approach the Korteweg-de Vries(KdV)equation is given here.According to such schemes,the full explicit difference scheme and the fun implicit one,an alternating segment explicit-implicit difference scheme for solving the KdV equation is constructed.The scheme is linear unconditionally stable by the analysis of linearization procedure,and is used directly on the parallel computer. The numerical experiments show that the method has high accuracy.     

Gas-kinetic numerical schemes for one- and two-dimensional inner flows

Several kinds of explicit and implicit finite-difference schemes directly solving the discretized velocity distribution functions are designed with precision of different orders by analyzing the inner characteristics of the gas-kinetic numerical algorithm for Boltzmann model equation. The peculiar flow phenomena and mechanism from various flow regimes are revealed in the numerical simulations of the unsteady Sod shock-tube problems and the two-dimensional channel flows with different Knudsen numbers. The numerical remainder-effects of the difference schemes are investigated aad analyzed based on the computed results. The ways of improving the computational efficiency of the gaskinetic numerical method and the computing principles of difference discretization are discussed.     

Developing shock-capturing difference methods

A new shock-capturing method is proposed which is based on upwind schemes and flux-vector splittings. Firstly, original upwind schemes are projected along characteristic directions. Secondly, the amplitudes of the characteristic decompositions are carefully controlled by limiters to prevent non-physical oscillations. Lastly, the schemes are converted into conservative forms, and the oscillation-free shock-capturing schemes are acquired. Two explicit upwind schemes (2nd-order and 3rd-order) and three compact upwind schemes (3rd-order, 5th-order and 7th-order) are modified by the method for hyperbolic systems and the modified schemes are checked on several one-dimensional and two-dimensional test cases. Some numerical solutions of the schemes are compared with those of a WENO scheme and a MP scheme as well as a compact-WENO scheme. The results show that the method with high order accuracy and high resolutions can capture shock waves smoothly.     

THE NUMERICAL SOLUTION OF A SINGULARLY PERTURBED PROBLEM FOR SEMILINEAR PARABOLIC DIFFERENTIAL EQUATION

The numerical solution of a singularly perturbed problem for the semilinear parabolicdifferential equation with parabolic boundary layers is discussed.A nonlinear two-leveldifference scheme is constructed on the special non-uniform grids.The uniform convergenceof this scheme is proved and some numerical examples are given.     

New discrete element models for elastoplastic problems

The discrete element method （DEM） has attractive features for problems with severe damages, but lack of theoretical basis for continua behavior especially for nonlinear behavior has seriously restricted its application. The present study proposes a new approach to developing the DEM as a general and robust technique for modeling the elastoplastic behavior of solid materials. New types of connective links between elements are proposed, the interelement parameters are theoretically determined based on the principle of energy equivalence and a yield criterion and a flow rule for DEM are given for describing nonlinear behavior of materials. Moreover, a numerical scheme, which can be applied to modeling the behavior of a continuum as well as the transformation from a continuum to a discontinuum, is obtained by introducing a fracture criterion and a contact model into the DEM. The elastoplastic stress wave propagations and the tensile failure process of a steel plate are simulated, and the numerical results agree well with those obtained from the finite element method （FEM） and corresponding experiment, and thus the accuracy and efficiency of the DEM scheme are demonstrated.     

UNIFIED COMPUTATION OF FLOW WITH COMPRESSIBLE AND INCOMPRESSIBLE FLUID BASED ON ROE''''S SCHEME

A unified numerical scheme for the solutions of the compressible and incompressible Navier-Stokes equations is investigated based on a time-derivative preconditioning algorithm. The primitive variables are pressure, velocities and temperature. The time integration scheme is used in conjunction with a finite volume discretization. The preconditioning is coupled with a high order implicit upwind scheme based on the definition of a Roe's type matrix. Computational capabilities are demonstrated through computations of high Mach number, middle Mach number, very low Mach number, and incompressible flow. It has also been demonstrated that the discontinuous surface in flow field can be captured for the implementation Roe's scheme.     

UNIFIED COMPUTATION OF FLOW WITH COMPRESSIBLE AND INCOMPRESSIBLE FLUID BASED ON ROE＇S SCHEME

A unified numerical scheme for the solutions of the compressible and incompressible Navier-Stokes equations is investigated based on a time-derivative preconditioning algorithm. The primitive variables are pressure, velocities and temperature. The time integration scheme is used in conjunction with a finite volume discretization. The preconditioning is coupled with a high order implicit upwind scheme based on the definition of a Roe＇s type matrix. Computational capabilities are demonstrated through computations of high Mach number, middle Mach number, very low Mach number, and incompressible flow. It has also been demonstrated that the discontinuous surface in flow field can be captured for the implementation Roe＇s scheme.     

New optimized flux difference schemes for improving high-order weighted compact nonlinear scheme with applications

To improve the spectral characteristics of the high-order weighted compact nonlinear scheme(WCNS), optimized flux difference schemes are proposed. The disadvantages in previous optimization routines, i.e., reducing formal orders, or extending stencil widths, are avoided in the new optimized schemes by utilizing fluxes from both cell-edges and cell-nodes. Optimizations are implemented with Fourier analysis for linear schemes and the approximate dispersion relation(ADR) for nonlinear schemes. Clas...     

Two exact solutions of the DPL non-Fourier heat conduction equation with special conditions

This paper presents two exact explicit solutions for the three dimensional dual-phase lag （DLP） heat conduction equation, during the derivation of which the method of trial and error and the authors＇ previous experiences are utilized. To the authors＇ knowledge, most solutions of 2D or 3D DPL models available in the literature are obtained by numerical methods, and there are few exact solutions up to now. The exact solutions in this paper can be used as benchmarks to validate numerical solutions and to develop numerical schemes, grid generation methods and so forth. In addition, they are of theoretical significance since they correspond to physically possible situations. The main goal of this paper is to obtain some possible exact explicit solutions of the dual-phase lag heat conduction equation as the benchmark solutions for computational heat transfer, rather than specific solutions for some given initial and boundary conditions. Therefore, the initial and boundary conditions are indeterminate before derivation and can be deduced from the solutions afterwards. Actually, all solutions given in this paper can be easily proven by substituting them into the governing equation.     

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.     

A FINITE DIFFERENCE METHOD FOR SIMULATING TRANSVERSE VIBRATIONS OF AN AXIALLY MOVING VISCOELASTIC STRING

A finite difference method is presented to simulate transverse vibrations of an axially moving string.By discretizing the governing equation and the equation of stress- strain relation at different frictional knots,two linear sparse finite difference equation systems are obtained.The two explicit difference schemes can be calculated alternatively, which make the computation much more efficient.The numerical method makes the nonlinear model easier to deal with and of truncation errors,O(Δt~2 Δx~2).It also shows quite good stability for small initial values.Numerical examples are presented to demonstrate the efficiency and the stability of the algorithm,and dynamic analysis of a viscoelastic string is given by using the numerical results.     

Theory and application of numerical simulation method of capillary force enhanced oil production

A kind of second-order implicit upwind fractional step finite difference methods are presented for the numerical simulation of coupled systems for enhanced(chemical)oil production with capillary force in the porous media.Some techniques,e.g.,the calculus of variations,the energy analysis method,the commutativity of the products of difference operators,the decomposition of high-order difference operators,and the theory of a priori estimate,are introduced.An optimal order error estimate in the l~2 norm is derived.The method is successfully used in the numerical simulation of the enhanced oil production in actual oilfields.The simulation results are satisfactory and interesting.     

Limit cycles and homoclinic orbits and their bifurcation of Bogdanov-Takens system

A quantitative analysis of limit cycles and homoclinic orbits, and the bifurcation curve for the Bogdanov-Takens system are discussed. The parameter incremental method for approximate analytical-expressions of these problems is given. These analytical-expressions of the limit cycle and homoclinic orbit are shown as the generalized harmonic functions by employing a time transformation. Curves of the parameters and the stability characteristic exponent of the limit cycle versus amplitude are drawn. Some of the limit cycles and homoclinic orbits phase portraits are plotted. The relationship curves of parameters μ and A with amplitude a and the bifurcation diagrams about the parameter are also given. The numerical accuracy of the calculation results is good.     

On dispersion-controlled principles for non-oscillatory shock-capturing schemes

The role of dispersions in the numerical solutions of hydrodynamic equation systems has been realized for long time. It is only during the last two decades that extensive studies on the dispersion-controlled dissipative (DCD) schemes were reported. The studies have demonstrated that this kind of the schemes is distinct from conventional dissipation-based schemes in which the dispersion term of the modified equation is not considered in scheme construction to avoid nonphysical oscillation occurring in shock wave simulations. The principle of the dispersion controlled aims at removing nonphysical oscillations by making use of dispersion characteristics instead of adding artificial viscosity to dissipate the oscillation as the conventional schemes do. Research progresses on the dispersioncontrolled principles are reviewed in this paper, including the exploration of the role of dispersions in numerical simulations, the development of the dispersion-controlled principles, efforts devoted to high-order dispersion-controlled dissipative schemes, the extension to both the finite volume and the finite element methods, scheme verification and solution validation, and comments on several aspects of the schemes from author‘s viewpoint.     

Incubation time fracture criterion for FEM simulations

The paper is discussing problems connected with embedment of the incubation time criterion for brittle fracture into finite element computational schemes.Incubation time fracture criterion is reviewed;practical questions of its numerical implementation are extensively discussed.Several examples of how the incubation time fracture criterion can be used as fracture condition in finite element computations are given.The examples include simulations of dynamic crack propagation and arrest,impact crater formation(i.e.fracture in initially intact media),spall fracture in plates,propagation of cracks in pipelines.Applicability of the approach to model initiation,development and arrest of dynamic fracture is claimed.     

DISSIPATION-BASED CONSISTENT RATE-DEPENDENT MODEL FOR CONCRETE

An energy-dissipation based viscoplastic consistency model is presented to describe the performance of concrete under dynamic loading. The development of plasticity is started with the thermodynamic hypotheses in order that the model may have a sound theoretical background. Independent hardening and softening and the rate dependence of concrete are described separately for tension and compression. A modified implicit backward Euler integration scheme is adopted for the numerical computation. Static and dynamic behavior of the material is illustrated with certain numerical examples at material point level and structural level, and compared with existing experimental data. Results validate the effectiveness of the model.     

The Difference Methods for the Solution of Singular-Perturbation for the Elliptic-Parabolic Differential Equation

In this paper it is discussed the difference method for the solution of singular perturbation pro-blems for the elliptic equations,involving small parameter in the higher derivatives.Asε=0 theoriginal equations are degenerated into the parabolic equations.Authors constructed special difference schemes by means of the boundary layer properties ofthe solutions of these problems as well as investigated the convergence of this scheme and asymptoticbehaviour of the solutions.Finally,a numerical example is given.     

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.     

Anisotropic rectangular nonconforming finite element analysis for Sobolev equations

An anisotropic rectangular nonconforming finite element method for solving the Sobolev equations is discussed under semi-discrete and full discrete schemes. The corresponding optimal convergence error estimates and superclose property are derived, which are the same as the traditional conforming finite elements. Furthermore, the global superconvergence is obtained using a post-processing technique. The numerical results show the validity of the theoretical analysis.     

PLASTIC LIMIT LOAD ANALYSIS OF DEFECTIVE PIPELINES

The integrity assessment of defective pipelines represents a practically important task of structural analysis and design in various technological areas, such as oil and gas industry, power plant engineering and chemical factories. An iterative algorithm is presented for the kinematic limit analysis of 3-D rigid-perfectly plastic bodies. A numerical path scheme for radial loading is adopted to deal with complex multi-loading systems. The numerical procedure has been applied to carry out the plastic collapse analysis of pipelines with part-through slot under internal pressure, bending moment and axial force. The effects of various shapes and sizes of part-through slots on the collapse loads of pipelines are systematically investigated and evaluated. Some typical failure modes corresponding to different configurations of slots and loading forms are studied.