The Boundedness of Calderón-Zygmund Operators by Wavelet Characterization

This article deals with the boundedness properties of Calderdn-Zygmund operators on Hardy spaces Hp（Rn）. We use wavelet characterization of H^P（R^n） to show that a Calderon-Zygmund operator T with T＊1 =0 is bounded on H6P（R^n）, n/n＋ε zju. edu. cn 〈 p 〈 1, where ε is the regular exponent of kernel of T. This approach can be applied to the boundedness of operators on certain Hardy spaces without atomic decomposition or molecular characterization.

Numerical Simulation of Moving Contact Lines with Surfactant by Immersed Boundary Method

In this paper,we present an immersed boundary method for simulating moving contact lines with surfactant.The governing equations are the incompressible Navier-Stokes equations with the usual mixture of Eulerian fluid variables and Lagrangian interfacial markers.The immersed boundary force has two components:one from the nonhomogeneous surface tension determined by the distribution of surfactant along the fluid interface,and the other from unbalanced Young’s force at the moving contact lines.An artificial tangential velocity has been added to the Lagrangian markers to ensure that the markers are uniformly distributed at all times.The corresponding modified surfactant equation is solved in a way such that the total surfactant mass is conserved.Numerical experiments including convergence analysis are carefully conducted.The effect of the surfactant on the motion of hydrophilic and hydrophobic drops are investigated in detail.

Numerical Study on Statistical Behaviors of Two-Dimensional Dry Foam

We investigate the statistical behaviors of two-dimensional dry foam using numerical simulations based on the immersed boundary(IB)method.We model the liquid phase of a foam as a thin elastic boundary with surface tension and the gas phase as a viscous incompressible fluid which can go through the liquid boundary.We present evidence of the existence of a limiting scaling state of the dry foam dynamics in which the asymptotic value ofμ_(2),the second moment of the distribution of the numbers of cell sides,lies in the range of 1.3±0.3.We also numerically ver-ify some well-known formulas derived in the dynamics of two-dimensional dry foam such as von Neumann relation,Lewis law,and Aboav-Weaire law.Our simulation re-sults are comparable to those of soap froth experiments and Potts model simulations.Furthermore,we investigate the statistical behaviors of two-dimensional dry foam in an oscillatory shear flow to show the applicability of our method to more general flow conditions.

A Coupled Immersed Interface and Level Set Method for Three-Dimensional Interfacial Flows with Insoluble Surfactant

In this paper,a numerical method is presented for simulating the 3D interfacial flows with insoluble surfactant.The numerical scheme consists of a 3D immersed interface method(IIM)for solving Stokes equations with jumps across the interface and a 3D level-set method for solving the surfactant convection-diffusion equation along a moving and deforming interface.The 3D IIM Poisson solver modifies the one in the literature by assuming that the jump conditions of the solution and the flux are implicitly given at the grid points in a small neighborhood of the interface.This assumption is convenient in conjunction with the level-set techniques.It allows standard Lagrangian interpolation for quantities at the projection points on the interface.The interface jump relations are re-derived accordingly.A novel rotational procedure is given to generate smooth local coordinate systems and make effective interpolation.Numerical examples demonstrate that the IIM Poisson solver and the Stokes solver achieve second-order accuracy.A 3D drop with insoluble surfactant under shear flow is investigated numerically by studying the influences of different physical parameters on the drop deformation.

Boundary Conditions for Limited Area Models Based on the Shallow Water Equations

A new set of boundary conditions has been derived by rigorousmethods for the shallow water equations in a limited domain.The aim of this article is to present these boundary conditions and to report on numerical simulations which have been performed using these boundary conditions.The new boundary conditions which are mildly dissipative let the waves move freely inside and outside the domain.The problems considered include a one-dimensional shallow water system with two layers of fluids and a two-dimensional inviscid shallow water system in a rectangle.

Modifying and Reducing Numerical Dissipation in A Two-Dimensional Central-Upwind Scheme

This study presents a modification of the central-upwind Kurganov scheme for approximating the solution of the 2D Euler equation.The prototype,extended from a 1D model,reduces substantially less dissipation than expected.The problem arises from over-restriction of some slope limiters,which keep slopes between interfaces of cells to be Total-Variation-Diminishing.This study reports the defect and presents a re-derived optimal formula.Numerical experiments highlight the significance of this formula,especially in long-time,large-scale simulations.

A Front-Tracking Method for Motion by Mean Curvature with Surfactant

In this paper,we present a finite difference method to track a network of curves whose motion is determined by mean curvature.To study the effect of inhomogeneous surface tension on the evolution of the network of curves,we include surfactant which can diffuse along the curves.The governing equations consist of one parabolic equation for the curve motion coupled with a convection-diffusion equation for the surfactant concentration along each curve.Our numerical method is based on a direct discretization of the governing equations which conserves the total surfactant mass in the curve network.Numerical experiments are carried out to examine the effects of inhomogeneous surface tension on the motion of the network,including the von Neumann law for cell growth in two space dimensions.

Preface Special Issue for SCPDE08: Numerical Methods and Analysis for PDEs and Inverse Problems

The Third International Conference on Scientific Computing and Partial Differential Equations(SCPDE)was held from December 8 to December 12,2008 at China Hong Kong Baptist University.It was a sequel to similar conferences held in Hong Kong region(2002 and 2005).The conference aims to promote research interests in scientific computation.In SCPDE 2008,there were 118 participants from seventeen countries and regions participated in the conference.The Programme included seventeen plenary addresses,thirty invited talks,twenty five contributed talks and seven poster presentations.