Shaoqiang Tang

A pseudo-spectral multiscale method: Interfacial conditions and coarse grid equations

In this paper, we propose a pseudo-spectral multiscale method for simulating complex systems with more than one spatial scale. Using a spectral decomposition, we split the displacement into its mean and fluctuation parts. Under the assumption of localized nonlinear fluctuations, we separate the domain into an MD (Molecular Dynamics) subdomain and an MC (MacrosCopic) subdomain. An interfacial condition is proposed across the two scales, in terms of a time history treatment. In the special case of a linear system, this is the first exact interfacial condition for multiscale computations. Meanwhile, we design coarse grid equations using a matching differential operator approach. The coarse grid discretization is of spectral accuracy. We do not use a handshaking region in this method. Instead, we define a coarse grid over the whole domain and reassign the coarse grid displacement in the MD subdomain with an average of the MD solution. To reduce the computational cost, we compute the dynamics of the coarse grid displacement and relate it to the mean displacement. Our method is therefore called a pseudo-spectral multiscale method. It allows one to reach high resolution by balancing the accuracy at the coarse grid with that at the interface. Numerical experiments in one- and two-space dimensions are presented to demonstrate the accuracy and the robustness of the method.

A finite difference approach with velocity interfacial conditions for multiscale computations of crystalline solids

We propose a class of velocity interfacial conditions and formulate a finite difference approach for multiscale computations of crystalline solids with relatively strong nonlinearity and large deformation. Full atomistic computations are performed in a selected small subdomain only. With a coarse grid cast over the whole domain and the coarse scale dynamics computed by finite difference schemes, we perform a fast average of the fine scale solution in the atomistic subdomain to force agreement between scales. During each coarse scale time step, we adopt a linear wave approximation around the interface, with the wave speed updated using the coarse grid information. We then develop a class of velocity interfacial conditions with different order of accuracy. The interfacial conditions are straightforward to formulate, easy to implement, and effective for reflection reduction in crystalline solids with strong nonlinearity. The nice features are demonstrated through numerical tests.

Dissipative Nonlinear Evolution Equations and Chaos

In this article we have studied the nonlinear interaction between ellipticity and dissipation in a set of model equations (1.1) and established the relation between this interaction and chaos. In addition to theoretical investigations, extensive numerical simulations with these equations have been made, and different routes to chaos have been found. The numerical studies have revealed the chaotic nature of the solutions.

A relaxation scheme for the hydrodynamic equations for semiconductors

In this paper, we shall study numerically the hydrodynamic model for semiconductor devices, particularly in a one-dimensional n+nn+ diode. By using a relaxation scheme, we explore the effects of various parameters, such as the low field mobility, device length, and lattice temperature. The effect of different types of boundary conditions is discussed. We also establish numerically the asymptotic limits of the hydrodynamic model towards the energy-transport and drift-diffusion models. This verifies the theoretical results in the literature.

Exact Boundary Condition for Semi-discretized Schrödinger Equation and Heat Equation in a Rectangular Domain

A convolution type exact/transparent boundary condition is proposed for simulating a semi-discretized linear Schrödinger equation on a rectangular computational domain. We calculate the kernel functions for a single source problem, and subsequently those over the rectangular domain. Approximate kernel functions are pre-computed numerically from discrete convolutionary equations. With a Crank–Nicolson scheme for time integration, the resulting approximate boundary conditions effectively suppress boundary reflections, and resolve the corner effect. The proposed boundary treatment, with a parameter modified, applies readily to a semi-discretized heat equation.

Accurate boundary treatment for transient Schrödinger equation under polar coordinates

Abstract A new local boundary condition is designed for the two dimensional Schrodinger equation under polar coordinates. Based on an approximate linear relation among the kernel functions for a free one-dimensional Schrodinger equation of a new variable, it takes a simple form of ordinary differential equation that relates the neighboring grid points. Numerical tests and comparisons demonstrate the effectiveness of the proposed boundary treatment.

Matching Boundary Conditions for Scalar Waves in Body-Centered-Cubic Lattices

Matching boundary conditions (MBCs) are proposed to treat scalar waves in the body-centered-cubic lattices. By matching the dispersion relation, we construct MBCs for normal incidence and incidence with an angle a. Multiplication of MBC operators then leads to multi-directional absorbing boundary conditions. The effec- tiveness are illustrated by the reflection coefficient analysis and wave packet tests. In particular, the designed M1M1 treats the scalar waves in a satisfactory manner. AMS subject classifications: 65Z05, 70-08

Homogenizing atomic dynamics by fractional differential equations

Abstract In this paper, we propose two ways to construct fractional differential equations (FDE) for approximating atomic chain dynamics. Taking harmonic chain as an example, we add a power function of fractional order to Taylor expansion of the dispersion relation, and determine the parameters by matching two selected wave numbers. This approximate function leads to an FDE after considering both directions for wave propagation. As an alternative, we consider the symbol of the force term, and approximate it by a similar function. It also induces an FDE. Both approaches produce excellent agreement with the harmonic chain dynamics. The accuracy may be improved by optimizing the selected wave numbers, or starting with higher order Taylor expansions. When resolved in the lattice constant, the resulting FDEs faithfully reproduce the lattice dynamics. When resolved in a coarse grid instead, they systematically generate homogenized algorithms. Numerical tests are performed to verify the proposed approaches. Moreover, FDEs are also constructed for diatomic chain and anharmonic lattice, to illustrate the generality of the proposed approaches.

ALmost EXact boundary conditions for transient Schrödinger-Poisson system

For the Schrodinger-Poisson system, we propose an ALmost EXact (ALEX) boundary condition to treat accurately the numerical boundaries. Being local in both space and time, the ALEX boundary conditions are demonstrated to be effective in suppressing spurious numerical reflections. Together with the Crank-Nicolson scheme, we simulate a resonant tunneling diode. The algorithm produces numerical results in excellent agreement with those in Mennemann et al. 1, yet at a much reduced complexity. Primary peaks in wave function profile appear as a consequence of quantum resonance, and should be considered in selecting the cut-off wave number for numerical simulations.