Nonstandard methods for the convective transport equation with nonlinear reactions

A new nonstandard Lagrangian method is constructed for the one-dimensional, transient convective transport equation with nonlinear reaction terms. An “exact” time-stepping scheme is developed with zero local truncation error with respect to time. The scheme is based on nonlocal treatment of nonlinear reactions, and when applied at each spatial grid point gives the new fully discrete numerical method. This approach leads to solutions free from the numerical instabilities that arise because of incorrect modeling of derivatives and nonlinear reaction terms. Algorithms are developed that preserve the properties of the numerical solution in the case of variable velocity fields by using nonuniform spatial grids. Effects of different interpolation techniques are examined and numerical results are presented to demonstrate the performance of the proposed new method. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 467–485, 1998     

A finite difference scheme for traveling wave solutions to Burgers equation

We construct a finite difference scheme for the ordinary differential equation describing the traveling wave solutions to the Burgers equation. This difference equation has the property that its solution can be calculated. Our procedure for determining this solution follows closely the analysis used to obtain the traveling wave solutions to the original ordinary differential equation. The finite difference scheme follows directly from application of the nonstandard rules proposed by Mickens. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 815–820, 1998     

Nylon 6 9 can crystallize with hydrogen bonding in two and in three interchain directions

Nylon 6 9 has been shown to have structures with interchain hydrogen bonds in both two and in three directions. Chain-folded lamellar crystals were studied using transmission electron microscopy and sedimented crystal mats and uniaxially oriented fibers studied by X-ray diffraction. The principal room-temperature structure shows the two characteristic (interchain) diffraction signals at spacings of 0.43 and 0.38 nm, typical of α-phase nylons; however, nylon 6 9 is unable to form the α-phase hydrogen-bonded sheets without serious distortion of the all-trans polymeric backbone. Our structure has c and c^* noncoincident and two directions of hydrogen bonding. Optimum hydrogen bonding can only occur if consecutive pairs of amide units alternate between two crystallographic planes. The salient features of our model offer a possible universal solution for the crystalline state of all odd–even nylons. The nylon 6 9 room-temperature structure has a C-centered monoclinic unit cell (β = 108°) with the hydrogen bonds along the C-face diagonals; this structure bears a similarity to that recently proposed for nylons 6 5 and X3. On heating nylon 6 9 lamellar crystals and fibers, the two characteristic diffraction signals converge and meet at 0.42 nm at the Brill temperature, T_B · T_B for nylon 6 9 lamellar crystals is slightly below the melting point (T_m), whereas T_B for nylon 6 9 fibers is ≅ 100°C below T_m. Above T_B, nylon 6 9 has a hexagonal unit cell; the alkane segments exist in a mobile phase and equivalent hydrogen bonds populate the three principal (hexagonal) directions. A structure with perturbed hexagonal symmetry, which bears a resemblance to the reported γ-phase for nylons, can be obtained by quenching from the crystalline growth phase (above T_B) to room temperature. We propose that this structure is a “quenched-in” perturbed form of the nylon 6 9 high-temperature hexagonal phase and has interchain hydrogen bonds in all three principal crystallographic directions. In this respect it differs importantly from the γ-phase models. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36: 1153–1165, 1998     

CALCULATION OF NAMBU MECHANICS

In this paper, a fundamental fact that Nambu mechanics is source free was proved.Based on this property, and via the idea of prolongation, finite dimensional Nambu systemwas prolonged to difference jet bundle. Structure preserving numerical methods of Nambuequations were established. Numerical experiments were presented at last to demonstrateadvantages of the structure preserving schemes.     

求解无粘可压Euler方程组的虚拟流方法

首先将三阶Godunov型半离散中心迎风格式推广到四阶,之后再将该新的四阶半离散中心迎风格式与Level Set方法以及虚拟流方法结合起来,成功地处理了非反应激波问题和多介质流中的爆轰间断问题。由于Level Set函数能隐式地追踪到界面的位置,而虚拟流的构造能隐式地捕捉到界面的边界条件,故而本文的方法可以很自然地推广到多维情况。     

HIGH-ORDER-ACCURATE SCHEMES FOR INCOMPRESSIBLE VISCOUS FLOW

We present new finite difference schemes for the incompressible Navier–Stokes equations. The schemes are based on two spatial differencing methods; one is fourth-order-accurate and the other is sixth-order accurate. The temporal differencing is based on backward differencing formulae. The schemes use non-staggered grids and satisfy regularity estimates, guaranteeing smoothness of the solutions. The schemes are computationally efficient. Computational results demonstrating the accuracy are presented. © 1997 by John Wiley & Sons, Ltd.     

Analysis and convergence of a MAC-like scheme for the generalized Stokes problem

We introduce a MAC-like scheme (a covolume method on rectangular grids) for approximating the generalized Stokes problem on an axiparallel domain. Two staggered grids are used in the derivation of the discretization. The velocity is approximated by conforming bilinears over rectangular elements, and the pressure by piecewise constants over macro-rectangular elements. The error in the velocity in the H¹ norm and the pressure in the L² norm are shown to be of first order, provided that the exact velocity is in H² and the exact pressure in H¹, and that the partition family of the domain is regular. © 1997 John Wiley & Sons, Inc.     

双曲型守恒律方程的熵稳定格式的一些讨论

本文讨论双曲型守恒律方程的熵稳定格式.对于给定的熵对,格式所满足的熵条件中的数值熵通量是不唯一的.Tadmor的充分条件可以唯一地确定标量方程的熵守恒通量,但不能唯一确定方程组的熵守恒通量,却可以给出方程组的空间一阶精度的熵守恒格式.也讨论了在熵守恒通量上添加数值粘性得到的显式熵稳定格式需要满足的条件及常见的时间离散对熵守恒和熵稳定的影响.     

VARIATIONS ON A THEME BY EULER

1.IntroductionAHallliltolliansystemofdifferentialequationsonRZnisgivedbyP~~H,(P,q),q=HP(P,q),(1)wherep=(pl,'.,P.),q=(ql,',q.)eR"arethegeneralizedcoordinatesandmolllentarespectivelyandH(P,q)istheellergyofthesystem.Thesystem(1)canberewrittenasthecompactf…     

Theoretical analysis of the Exponential Transversal Method of Lines for the diffusion equation

A new approximate technique to solve the diffusion equation, called the Exponential Transversal Method of Lines (ETMOL), utilizes an exponential variation of the dependent variable to improve accuracy in the evaluation of the time derivative. Campo and Salazar have implemented this method in a wide range of heat/mass transfer applications and have obtained surprisingly good numerical results. In this article, we study the theoretical properties of ETMOL in depth. In particular, consistency, stability, and convergence are established within the framework of the heat/mass diffusion equation. In most practical applications, the new method presents a very reduced truncation error in time, and its different versions are proven to be unconditionally stable in the Fourier sense. Convergence of the approximate solutions have then been established. The theory is corroborated by several analytical/numerical experiments that pose different levels of complexity. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 30–41, 2000