In this paper we survey recent progress in symplectic algorithms for use in quantum systems in the following topics: Symplectic schemes for solving Hamiltonian systems; Classical trajectories of diatomic systems, model molecule A₂B, Hydrogen ion H₂⁺ and elementary atmospheric reaction N(⁴S)+O₂(X³Σ₈⁻)→NO(X²Π)+O(³P) calculated by means of Runge-Kutta methods and symplectic methods; the classical dissociation of the HF molecule and classical dynamics of H₂⁺ in an intense laser field; the symplectic form and symplectic-scheme shooting method for the time-independent Schrödinger equation; the computation of continuum eigenfunction of the Schrödinger equation; asymptotic boundary conditions for solving the time-dependent Schrödinger equation of an atom in an intense laser field; symplectic discretization based on asymptotic boundary condition and the numerical eigenfunction expansion; and applications in computing multi-photon ionization, above-threshold ionization, Rabbi oscillation and high-order harmonic generation of laser-atom interaction.

An efficient and accurate spectral method is presented for scattering problems with rough surfaces. A probabilistic framework is adopted by modeling the surface roughness as random process. An improved boundary perturbation technique is employed to transform the original Helmholtz equation in a random domain into a stochastic Helmholtz equation in a fixed domain. The generalized polynomial chaos (gPC) is then used to discretize the random space; and a Fourier-Legendre method to discretize the physical space. These result in a highly efficient and accurate spectral algorithm for acoustic scattering from rough surfaces. Numerical examples are presented to illustrate the accuracy and efficiency of the present algorithm.

The surface diffusion of an axi-symmetric solid, a whisker, subject to applied uniaxial stress, is studied numerically based on a new boundary integral formulation for periodic stress configurations. An efficient semi-implicit time-stepping scheme is developed to treat the serve stiffness due to high-order derivatives. When the initial perturbation is small the effect of the stress on the motion of the whisker is found to agree with the linear stability analysis. Numerical simulations of a fully nonlinear case are also presented, and a potential break-up of the whisker is observed.

We discuss the basic concept of compartmental modelling in pharmacokinetics and demonstrate that all the solutions admitted by multi-compartment models of classical pharmacokinetics are expressed as linear combinations of exponential functions of time. This lends itself to data analysis that depends on fitting exponential functions to finite size sets. A mathematical method developed a long time ago to deal with this type of problem is called Prony's method. We discuss the usefulness of this method in pharmacokinetic modeling and apply it to a particular data set obtained for the drug mibefradil. In spite of the method's power in dealing with well-behaved data sets, we indicate the existence of severe limitations since real concentration curves coming from pharmacokinetic data are seldom purely exponential.

Numerical simulations have been carried out for laminar sinusoidal pulsating flow in a tube with smooth single constriction. A second-order finite volume method has been developed to solve the fluid flow governing equations on a nonstaggered non-orthogonal grid. The effects of the Reynolds number, the Womersley number, the pulsatile amplitude, the constriction ratio and the constriction length on fluid flow in constricted tube will be investigated. It will be demonstrated that the dynamic nature of the pulsating flow greatly depends on the frequency of the flow changes. It is observed that the peak wall vorticity seems to increase with the increase of Reynolds number, the pulsating amplitude and the constriction ratio. The peak values of instantaneous wall vorticity are not greatly affected by the variation of Womersly number. The constriction length does not put a significant impact on the flow instantaneous streamline behaviors compared with other parameters. However, the peak wall vorticity increases monotonically with the decrease of the constriction length.

This paper introduces an extension of the time-splitting spectral (TSSP) method for solving a general model of three-wave optical interactions, which typically arises from nonlinear optics, when the transmission media has competing quadratic and cubic nonlinearities. The key idea is to formulate the terms related to quadratic and cubic nonlinearities into a Hermitian matrix in a proper way, which allows us to develop an explicit and unconditionally stable numerical method for the problem. Furthermore, the method is spectral accurate in transverse coordinates and second-order accurate in propagation direction, is time reversible and time transverse invariant, and conserves the total wave energy (or power or the norm of the solutions) in discretized level. Numerical examples are presented to demonstrate the efficiency and high resolution of the method. Finally the method is applied to study dynamics and interactions between three-wave solitons and continuous waves in media with competing quadratic and cubic nonlinearities in one dimension (1D) and 2D.

We study central-upwind schemes for systems of hyperbolic conservation laws, recently introduced in [13]. Similarly to staggered non-oscillatory central schemes, these schemes are central Godunov-type projection-evolution methods that enjoy the advantages of high resolution, simplicity, universality and robustness. At the same time, the central-upwind framework allows one to decrease a relatively large amount of numerical dissipation present at the staggered central schemes. In this paper, we present a modification of the one-dimensional fully- and semi-discrete central-upwind schemes, in which the numerical dissipation is reduced even further. The goal is achieved by a more accurate projection of the evolved quantities onto the original grid. In the semi-discrete case, the reduction of dissipation procedure leads to a new, less dissipative numerical flux. We also extend the new semi-discrete scheme to the two-dimensional case via the rigorous, genuinely multidimensional derivation. The new semi-discrete schemes are tested on a number of numerical examples, where one can observe an improved resolution, especially of the contact waves.

Kinetic Monte Carlo (KMC) is a stochastic model used to simulate crystal growth. However, most KMC models rely on a pre-defined lattice that neglects dislocations, lattice mismatch and strain effects. In this paper, we investigate the use of a 3D off-lattice KMC algorithm. We test this method by investigating impurity diffusion in a strained FCC nanowire. While faster than a molecular dynamics simulation, the most general implementation of off-lattice KMC is much slower than a lattice-based algorithm. An improved procedure is achieved for weakly strained systems by precomputing approximate saddle point locations based on unstrained lattice structures. In this way, one gives up some of the flexibility of the general method to restore some of the computational speed of lattice-based KMC. In addition to providing an alternative approach to nano-materials simulation, this type of simulation will be useful for testing and calibrating methods that seek to parameterize the variation in the transition rates for lattice-based KMC using continuum modeling.