Shaozhong Deng

Numerical study of light propagation via whispering gallery modes in microcylinder coupled resonator optical waveguides.

By using a discontinuous spectral element method, we analyze evanescent wave coupling of whispering gallery modes (WGMs) in microcylinder coupled resonator optical waveguides (CROWs). We demonstrate successful light propagation by WGMs through a chain of coupled cylinder resonators, and that the speed of such propagation is strongly dependent on the inter-resonator gap sizes. Our simulations also show that light propagates slower byWGMs with bigger azimuthal numbers than by those with smaller azimuthal numbers. On the other hand, the light propagation by WGMs of the same azimuthal number appears to have the same speed in CROWs regardless of the size and the material of the resonators, indicating that the tail (the part of a WGM outside the resonator) determines inter-resonator coupling strength.

Extending the fast multipole method to charges inside or outside a dielectric sphere

In this paper, we propose a novel method to extend the fast multipole method (FMM) to calculate the electrostatic potential due to charges inside or outside a dielectric sphere. The key result which allows such an extension is the construction of a small number (two for a 10^-^2 relative error in reaction potentials inside the sphere) of image point charges for source point charges inside or outside the dielectric sphere. Numerical results validate the accuracy and high efficiency of the resulting algorithm.

Three-dimensional elliptic solvers for interface problems and applications

Second-order accurate elliptic solvers using Cartesian grids are presented for three-dimensional interface problems in which the coefficients, the source term, the solution and its normal flux may be discontinuous across an interface. One of our methods is designed for general interface problems with variable but discontinuous coefficient. The scheme preserves the discrete maximum principle using constrained optimization techniques. An algebraic multigrid solver is applied to solve the discrete system. The second method is designed for interface problems with piecewise constant coefficient. The method is based on the fast immersed interface method and a fast 3D Poisson solver. The second method has been modified to solve Helmholtz/Poisson equations on irregular domains. An application of our method to an inverse interface problem of shape identification is also presented. In this application, the level set method is applied to find the unknown surface iteratively.

An upwinding embedded boundary method for Maxwell's equations in media with material interfaces: 2D case

We propose a new upwinding embedded boundary method to solve time dependent Maxwells equations in media with material interfaces. A global second order finite difference method is obtained by combining central difference schemes away from the interfaces and upwinding technique with jump conditions near the interfaces. The proposed finite difference method allows time step based on a uniform mesh independent of the locations and shapes of the interfaces. Moreover, the scheme is simple to implement in multidimensional cases. Numerical tests of wave equations with various types of material interfaces and electromagnetic scattering of 2D cylinders confirm the stability, uniform accuracy and ease of implementation of the method.

An image-based reaction field method for electrostatic interactions in molecular dynamics simulations of aqueous solutions.

In this paper, a new solvation model is proposed for simulations of biomolecules in aqueous solutions that combines the strengths of explicit and implicit solvent representations. Solute molecules are placed in a spherical cavity filled with explicit water, thus providing microscopic detail where it is most needed. Solvent outside of the cavity is modeled as a dielectric continuum whose effect on the solute is treated through the reaction field corrections. With this explicit/implicit model, the electrostatic potential represents a solute molecule in an infinite bath of solvent, thus avoiding unphysical interactions between periodic images of the solute commonly used in the lattice-sum explicit solvent simulations. For improved computational efficiency, our model employs an accurate and efficient multiple-image charge method to compute reaction fields together with the fast multipole method for the direct Coulomb interactions. To minimize the surface effects, periodic boundary conditions are employed for nonelectrostatic interactions. The proposed model is applied to study liquid water. The effect of model parameters, which include the size of the cavity, the number of image charges used to compute reaction field, and the thickness of the buffer layer, is investigated in comparison with the particle-mesh Ewald simulations as a reference. An optimal set of parameters is obtained that allows for a faithful representation of many structural, dielectric, and dynamic properties of the simulated water, while maintaining manageable computational cost. With controlled and adjustable accuracy of the multiple-image charge representation of the reaction field, it is concluded that the employed model achieves convergence with only one image charge in the case of pure water. Future applications to pKa calculations, conformational sampling of solvated biomolecules and electrolyte solutions are briefly discussed.

Ionic solvation studied by image-charge reaction field method

In a preceding paper [J. Chem. Phys. 131, 154103 (2009)], we introduced a new, hybrid explicit/implicit method to treat electrostatic interactions in computer simulations, and tested its performance for liquid water. In this paper, we report further tests of this method, termed the image-charge solvation model (ICSM), in simulations of ions solvated in water. We find that our model can faithfully reproduce known solvation properties of sodium and chloride ions. The charging free energy of a single sodium ion is in excellent agreement with the estimates by other electrostatics methods, while offering much lower finite-size errors. Similarly, the potentials of mean force computed for Na-Cl, Na-Na, and Cl-Cl pairs closely reproduce those reported previously. Collectively, our results demonstrate the superior accuracy of the proposed ICSM method for simulations of mixed media.

Extending the fast multipole method for charges inside a dielectric sphere in an ionic solvent: High-order image approximations for reaction fields

As a sequel to our previous paper on extending the Fast Multipole Method (FMM) for charges inside a dielectric sphere [J. Comput. Phys. 223 (2007) 846-864], this paper further extends the FMM to the electrostatic calculation for charges inside a dielectric sphere immersed in an ionic solvent, a scenery with more relevance in biological applications. The key findings include two fourth-order multiple discrete image approximations in terms of u=λa to the reaction field induced by the ionic solvent, provided that u=λa < 1 where λ is the inverse Debye screening length of the ionic solvent and a is the radius of the dielectric sphere. A 10(-4) relative accuracy in the reaction field of a source charge within the sphere can be achieved with only 3-4 point image charges. Together with the image charges, the FMM can be used to speed up the calculation of electrostatic interactions of charges in a dielectric sphere immersed in an ionic solvent.

A robust numerical method for self-polarization energy of spherical quantum dots with finite confinement barriers

By utilizing a novel three-layer dielectric model for the interface between a spherical quantum dot and the surrounding matrix, a robust numerical method for calculating the self-polarization energy of a spherical quantum dot with a finite confinement barrier is presented in this paper. The proposed numerical method can not only overcome the inherent mathematical divergence in the self-polarization energy which arises for the simplest and most widely used step-like model of the dielectric interface, but also completely eliminate the potential numerical divergence which may occur in the Bolcatto-Proettos formula [J. Phys.: Condens. Matter 13, 319-334 (2001)], an approximation method commonly employed for more realistic three-layer dielectric models such as the linear and the cosine-like models frequently mentioned in the literature. Numerical experiments have demonstrated the convergence of the proposed numerical method as the number of the steps used to discretize the translation layer in a three-layer model goes to infinity, an important property that the Bolcatto-Proettos formula appears not necessarily to possess.

Image charge approximations of reaction fields in solvents with arbitrary ionic strength

In this paper, we report a new development of image charge approximations of reaction fields for a charge inside a dielectric spherical cavity immersed in an ionic solvent with arbitrary ionic strength. This new development removes the requirement of low ionic strength of the solvent in a previous result [S. Deng, W. Cai, Extending the fast multipole method for charges inside a dielectric sphere in an ionic solvent: high-order image approximations for reaction fields, J. Comput. Phys. 227 (2007) 1246-1266], thus extending the applicability of the image charge approximations of reaction fields in the modeling of bio-molecular solvation.

A comparable study of image approximations to the reaction field

The recently developed high-order accurate multiple image approximation to the reaction field for a charge inside a dielectric sphere [J. Comput. Phys., 223 (2007) 846-864] is compared favorably to other commonly employed reaction field schemes. These methods are of particular interest because they are useful in the study of biological macromolecules by the Monte Carlo and Molecular Dynamics methods.