Existence and comparison results of solutions for nonlinear random integral and differential equations relative to weak topology

In this paper, we prove several existence theorems of random solutions to nonlinear random Volterra integral equations under the weak topology of Banach spaces. Then, as applications, we obtain the existence theorems of weak random solutions to random differential equations. Existence of extremal random solutions and a random comparison theorem for these random equations are also obtained. Our theorems improve and extend the corresponding results in [4,5,10,11,12]. Projects supported by the Science Fund of the Chinese Academy of Sciences.     

Finite element computations of viscoelastic two-phase flows using local projection stabilization

A three-field local projection stabilized (LPS) finite element method is developed for computations of a three-dimensional axisymmetric buoyancy driven liquid drop rising in a liquid column where one of the liquid is viscoelastic. The two-phase flow is described by the time-dependent incompressible Navier-Stokes equations, whereas the viscoelasticity is modeled by the Giesekus constitutive equation in a time-dependent domain. The arbitrary Lagrangian-Eulerian (ALE) formulation with finite elements is used to solve the governing equations in the time-dependent domain. Interface-resolved moving meshes in ALE allows to incorporate the interfacial tension force and jumps in the material parameters accurately. A one-level LPS based on an enriched approximation space and a discontinuous projection space is used to stabilize the numerical scheme. A comprehensive numerical investigation is performed for a Newtonian drop rising in a viscoelastic fluid column and a viscoelastic drop rising in a Newtonian fluid column. The influence of the viscosity ratio, Newtonian solvent ratio, Giesekus mobility factor, and the Eötvös number on the drop dynamics are analyzed. The numerical study shows that beyond a critical Capillary number, a Newtonian drop rising in a viscoelastic fluid column experiences an extended trailing edge with a cusp-like shape and also exhibits a negative wake phenomena. However, a viscoelastic drop rising in a Newtonian fluid column develops an indentation around the rear stagnation point with a dimpled shape.     

A stabilized incremental projection scheme for the incompressible Navier–Stokes equations

It is well known that any spatial discretization of the saddle‐point Stokes problem should satisfy the Ladyzhenskaya–Brezzi–Babuska (LBB) stability condition in order to prevent the appearance of spurious pressure modes. Particularly, if an equal‐order approximation is applied, the Schur complement (or, as called some times, the Uzawa matrix) of the pressure system has a non‐trivial null space that gives rise to such modes. An idea in the past was that all the schemes that solve a Poisson equation for the pressure rather than the Uzawa pressure equation (splitting/projection methods) should overcome this difficulty; this idea was wrong. There is numerical evidence that at least the so‐called incremental projection scheme still suffers from spurious pressure oscillations if an equal‐order approximation is applied. The present paper tries to distinguish which projection requires LBB‐compliant approximation and which does not. Moreover, a stabilized version of the incremental projection scheme is derived. Proper bounds for the stabilization parameter are also given. The numerical results show that the stabilized scheme does indeed achieve second‐order accuracy and does not produce spurious (node to node) pressure oscillations. Copyright © 2001 John Wiley & Sons, Ltd.     

Efficient augmented Lagrangian‐type preconditioning for the Oseen problem using Grad‐Div stabilization

Efficient preconditioning for Oseen‐type problems is an active research topic. We present a novel approach leveraging stabilization for inf‐sup stable discretizations. The Grad‐Div stabilization shares the algebraic properties with an augmented Lagrangian‐type term. Both simplify the approximation of the Schur complement, especially in the convection‐dominated case. We exploit this for the construction of the preconditioner. Solving the discretized Oseen problem with an iterative Krylov‐type method shows that the outer iteration numbers are retained independent of mesh size, viscosity, and finite element order. Thus, the preconditioner is very competitive. Copyright © 2012 John Wiley & Sons, Ltd.     

Local projection stabilized finite element method for Navier-Stokes equations

This paper extends the results of Matthies, Skrzypacz, and Tubiska for the Oseen problem to the Navier-Stokes problem. For the stationary incompressible Navier- Stokes equations, a local projection stabilized finite element scheme is proposed. The scheme overcomes convection domination and improves the restrictive inf-sup condition. It not only is a two-level approach but also is adaptive for pairs of spaces defined on the same mesh. Using the approximation and projection spaces defined on the same mesh, the scheme leads to much more compact stencils than other two-level approaches. On the same mesh, besides the class of local projection stabilization by enriching the approximation spaces, two new classes of local projection stabilization of the approximation spaces are derived, which do not need to be enriched by bubble functions. Based on a special interpolation, the stability and optimal prior error estimates are shown. Numerical results agree with some benchmark solutions and theoretical analysis very well.     

A new streamline diffusion finite element method for the generalized Oseen problem

This paper aims to present a new streamline diffusion method with low order rectangular Bernardi-Raugel elements to solve the generalized Oseen equations. With the help of the Bramble-Hilbert lemma, the optimal errors of the velocity and pressure are estimated, which are independent of the considered parameter ε. With an interpolation postprocessing approach, the superconvergent error of the pressure is obtained. Finally,a numerical experiment is carried out to confirm the theoretical results.     

Lie group integration for constrained generalized Hamiltonian system with dissipation by projection method

For the constrained generalized Hamiltonian system with dissipation, by introducing Lagrange multiplier and using projection technique, the Lie group integration method was presented, which can preserve the inherent structure of dynamic system and the constraintinvariant. Firstly, the constrained generalized Hamiltonian system with dissipative was converted to the non-constraint generalized Hamiltonian system, then Lie group integration algorithm for the non-constraint generalized Hamiltonian system was discussed, finally the projection method for generalized Hamiltonian system with constraint was given. It is found that the constraint invariant is ensured by projection technique, and after introducing Lagrange multiplier the Lie group character of the dynamic system can‘ t be destroyed while projecting to the constraint manifold. The discussion is restricted to the case of holonomic constraint. A presented numerical example shows the effectiveness of the method.     

Periodicity and strict oscillation for generalized lyness equations

ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｄｅｌａｙｄｉｆｆｅｒｅｎｃｅｅｑｕａｔｉｏｎｘｎ 1=ｘｎ(ａ ｂｘｎ)ｘｎ- 1,　　ｎ=0 ,1 ,2 ,… ,(1 )ｗｈｅｒｅａ ,ｂ∈ [0 ,∞ )ｗｉｔｈａ ｂ>0 (2 )ａｎｄｗｈｅｒｅｔｈｅｉｎｉｔｉａｌｖａｌｕｅｓｘ- 1ａｎｄｘ0 ａｒｅａｒｂｉｔｒａｒｙｐｏｓｉｔｉｖｅｎｕｍｂｅｒｓ.Ｅｑ .(1 )ｉｓｒｅｇａｒｄｅｄａｓａｇｅｎｅｒａｌｉｚｅｄＬｙｎｅｓｓｅｑｕａｔｉｏｎｂｙＧ .Ｌａｄａｓｉｎ [1 ] .Ｏｂｖｉｏｕｓｌｙ ,ｕｎｄｅｒｃｏｎｄｉｔｉｏｎ (2 ) ,…     

Explicit solitary-wave solutions to generalized Pochhammer-Chree equations

ｎｔｒｏｄｕｃｔｉｏｎＰｏｃｈｈａｍｍｅｒ＿Ｃｈｒｅｅｑｕａｔｉｏｎ（ＰＣｅｑｕａｔｉｏｎｉｎｓｈｏｒｔ）ｕｔ－ｕｔｘｘ－ｕｘｘ－１ｐ（ｕｐ）ｘｘ＝０,（１）ｉｓｕｓｅｄｔｏｄｅｓｃｒｉｂｅｔｈｅｐｒｏｐａｇａｔｉｏｎｏｆｌｏｎｇｉｔｕｄｉｎａｌｄｅ...     

BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS FOR GENERALIZED DRINFELD-SOKOLOV EQUATIONS

Ansatz method and the theory of dynamical systems are used to study the traveling wave solutions for the generalized Drinfeld-Sokolov equations. Under two groups of the parametric conditions, more solitary wave solutions, kink and anti-kink wave solutions and periodic wave solutions are obtained. Exact explicit parametric representations of these travelling wave solutions are given.     

The solution for the generalized riccati algebraic equations of linear equality constraint system

Based on the dynamic equation, the performance functional and the system constraint equation of time-invariant discrete LQ control problem, the generalized Riccati equations of linear equality constraint system are obtained according to the minimum principle, then a deep discussion about the above equations is given, and finally numerical example is shown in this paper. Project supported by the National Natural Science Youth Foundation of China     

多体系统动力学方程违约修正的数值计算方法

多体系统动力学方程为微分代数方程,一般将其转化成常微分方程组进行数值计算,在数值积分的过程中约束方程的违约会逐渐增大.本文对具有完整、定常约束的多体系统,在修改的带乘子Lagrange正则形式的方程的基础上,根据Baumgarte提出的违约修正的方法,给出了一种多体系统微分代数方程违约修正法和系统的动力学方程的矩阵表达式.通过对曲柄-滑块机构的数值仿真,计算结果表明本文给出的方法在计算精度和计算效率上好于Baumgarte提出的两种违约修正的方法.     

Potential symmetries and conservation laws for generalized quasilinear hyperbolic equations

Based on the Lie group method, the potential symmetries and invariant solutions for generalized quasilinear hyperbolic equations are studied. To obtain the invariant solutions in an explicit form, the physically interesting situations with potential symmetries are focused on, and the conservation laws for these equations in three physically interesting cases are found by using the partial Lagrangian approach.     

Exponential stability of stochastic generalized porous media equations with jump

Stochastic generalized porous media equation with jump is considered. The aim is to show the moment exponential stability and the almost certain exponential stability of the stochastic equation.     

Analysis of chebyshev pseudospectral method for multi-dimensional generalized srlw equations

IntroductionAsymmetricregularizedlongwaveequation (SRLWE) 2 x2 -1 u t = x ρ+ 12 u2 ,　 ρ t+ u x=0 (1 )hasbeeninvestigatedinRef.[1 ] .Thesystem (1 )ofequationsisshowntodescribeweaklynonlinearion_acousticwaveandspace_chargewaves.Thehuperbolicsecantsquaredsolitarywaves ,thefourconservationlaws,andsomenumericalresultshavebeenobtainedinRef.[1 ] .Obviously ,eliminatingρin (1 ) ,weobtainaclassofregularizedlongwaveequationutt-uxx+ 12 u2xt-uxxtt =0 . (2 )TheSRLWequationisexplicitlysymmetric…     

Superconvergence of nonconforming finite element penalty scheme for Stokes problem using L 2 projection method

A modified penalty scheme is discussed for solving the Stokes problem with the Crouzeix-Raviart type nonconforming linear triangular finite element. By the L ² projection method, the superconvergence results for the velocity and pressure are obtained with a penalty parameter larger than that of the classical penalty scheme. The numerical experiments are carried out to confirm the theoretical results.     

The addendum for the generalized d'Alembert equations of motion

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SELF-ADAPTIVE STRATEGY FOR ONE-DIMENSIONAL FINITE ELEMENT METHOD BASED ON ELEMENT ENERGY PROJECTION METHOD

Based on the newly-developed element energy projection (EEP) method for computation of super-convergent results in one-dimensional finite element method (FEM), the task of self-adaptive FEM analysis was converted into the task of adaptive piecewise polynomial interpolation. As a result, a satisfactory FEM mesh can be obtained, and further FEM analysis on this mesh would immediately produce an FEM solution which usually satisfies the user specified error tolerance. Even though the error tolerance was not completely satisfied, one or two steps of further local refinements would be sufficient. This strategy was found to be very simple, rapid, cheap and efficient. Taking the elliptical ordinary differential equation of second order as the model problem, the fundamental idea, implementation strategy and detailed algorithm are described. Representative numerical examples are given to show the effectiveness and reliability of the proposed approach.     

Discontinuous element pressure gradient stabilizations for compressible Navier-Stokes equations based on local projections

A pressure gradient discontinuous finite element formulation for the compressible Navier-Stokes equations is derived based on local projections. The resulting finite element formulation is stable and uniquely solvable without requiring a B-B stability condition. An error estimate is Obtained.     

The generalized Nielsen's equations for controllable variable-mass systems

In this paper, Nielsen's form of Gauss's principle for controllable variable-mass systems is established. By this principle, the generalized Nielsen's Equations are extended to controllable variable-mass systems. The equations in terms of quasi-coordinates are obtained. The equivalence between the generalized Nielsen's equations and Appell's equations is demonstrated.