Novel Conservative Methods for Schrödinger Equations with Variable Coefficients over Long Time

In this paper,we propose a wavelet collocation splitting(WCS)method,and a Fourier pseudospectral splitting(FPSS)method as comparison,for solving onedimensional and two-dimensional Schrödinger equations with variable coefficients in quantum mechanics.The two methods can preserve the intrinsic properties of original problems as much as possible.The splitting technique increases the computational efficiency.Meanwhile,the error estimation and some conservative properties are investigated.It is proved to preserve the charge conservation exactly.The global energy and momentum conservation laws can be preserved under several conditions.Numerical experiments are conducted during long time computations to show the performances of the proposed methods and verify the theoretical analysis.

THE GENERIC PROPERTIES OF SOLUTIONS OF A CONSERVATION LAW

In this paper we answer all the questions about the conjectures of Glimm and Lax on genericproperties of solutions. We prove that the discotinuous points of almost every solution with L∞,bounded varistion or contrinuous data are dense in the upper half-plane minus the closure of the setof central simple waves. It is also proved that if the equation is analytic,then the solutions withpiecewise analytic data are piecewise analytic,and the shock curves are also piecewise analytic. Wedisprove the conjecture which claims that almost every solution with C^k data is 'bad' enough, and provethat every solution with C^k data possesses nice propetied, i.e. when k≥4 the generic property ofsolutions is piecewise C^k,and hence is 'good' enough.For the proof of the generic property withC^k (k≥4) data, the idea of transversality in the theory of singular points is essential.

Numerical Simulation of Free Surface by an Area-Preserving Level Set Method

We apply in this study an area preserving level set method to simulate gas/water interface flow.For the sake of accuracy,the spatial derivative terms in the equations of motion for an incompressible fluid flow are approximated by the fifthorder accurate upwinding combined compact difference(UCCD)scheme.This scheme development employs two coupled equations to calculate the first-and second-order derivative terms in the momentum equations.For accurately predicting the level set value,the interface tracking scheme is also developed to minimize phase error of the first-order derivative term shown in the pure advection equation.For the purpose of retaining the long-term accurate Hamiltonian in the advection equation for the level set function,the time derivative term is discretized by the sixth-order accurate symplectic Runge-Kutta scheme.Also,to keep as a distance function for ensuring the front having a finite thickness for all time,the re-initialization equation is used.For the verification of the optimized UCCD scheme for the pure advection equation,two benchmark problems have been chosen to investigate in this study.The level set method with excellent area conservation property proposed for capturing the interface in incompressible fluid flows is also verified by solving the dam-break,Rayleigh-Taylor instability,two-bubble rising in water,and droplet falling problems.