Zhongyi Huang

Matching conditions in atomistic-continuum modeling of materials.

A new class of matching conditions between the atomistic and continuum regions is presented for the multiscale modeling of crystals. They ensure the accurate passage of large scale information between the atomistic and continuum regions and at the same time minimize the reflection of phonons at the interface. These matching conditions can be made adaptive if we choose appropriate weight functions. Applications to dislocation dynamics and friction between two-dimensional atomically flat crystal surfaces are described.

A dynamic atomistic-continuum method for the simulation of crystalline materials

We present a coupled atomistic-continuum method for the modeling of defects and interface dynamics in crystalline materials. The method uses atomistic models such as molecular dynamics near defects and interfaces, and continuum models away from defects and interfaces. We propose a new class of matching conditions between the atomistic and the continuum regions. These conditions ensure the accurate passage of large-scale information between the atomistic and the continuum regions and at the same time minimize the reflection of phonons at the atomistic-continuum interface. They can be made adaptive by choosing appropriate weight functions. We present applications to dislocation dynamics, friction between two-dimensional crystal surfaces, and fracture dynamics. We compare results of the coupled method and of the detailed atomistic model.

A class of artificial boundary conditions for heat equation in unbounded domains

Abstract In this paper, the numerical solutions of three problems of heat equation on unbounded domains are considered. For each problem, we introduce an artificial boundary Γ to make the computational domain finite. On the artificial boundary Γ, we propose an exact artificial boundary condition to reduce the original problem to an initial-boundary problem of heat equation on the finite computational domain, which is equivalent to the original problem. Then the finite difference method and finite element method are used to solve the reduced problem on the finite computational domain. In the end of this paper, three numerical examples show the feasibility and effectiveness of the method given in this paper.

A time-splitting spectral scheme for the Maxwell-Dirac system

We present a time-splitting spectral scheme for the Maxwell-Dirac system and similar time-splitting methods for the corresponding asymptotic problems in the semi-classical and the non-relativistic regimes. The scheme for the Maxwell-Dirac system conserves the Lorentz gauge condition is unconditionally stable and highly efficient as our numerical examples show. In particular, we focus in our examples on the creation of positronic modes in the semi-classical regime and on the electron-positron interaction in the non-relativistic regime. Furthermore, in the non-relativistic regime, our numerical method exhibits uniform convergence in the small parameter @d, which is the ratio of the characteristic speed and the speed of light.

Exact and approximating boundary conditions for the parabolic problems on unbounded domains

Abstract In this paper, the numerical solutions of the problems of heat equation in two dimensions on unbounded domains are considered. For a given problem, we introduce an artificial boundary Γ to finite the computational domain. On the artificial boundary Γ, we propose an exact boundary condition to reduce the given problem to an initial-boundary problem of heat equation on the finite computational domain, which is equivalent to the original problem. Furthermore, a series of approximating artificial boundary conditions is given. Then the finite difference method and finite element method are used to solve the reduced problem on the finite computational domain. Finally, the numerical examples show the feasibility and effectiveness of the method given in this paper.

A Tailored Finite Point Method for a Singular Perturbation Problem on an Unbounded Domain

In this paper, we propose a tailored-finite-point method for a kind of singular perturbation problems in unbounded domains. First, we use the artificial boundary method (Han in Frontiers and Prospects of Contemporary Applied Mathematics, [2005]) to reduce the original problem to a problem on bounded computational domain. Then we propose a new approach to construct a discrete scheme for the reduced problem, where our finite point method has been tailored to some particular properties or solutions of the problem. From the numerical results, we find that our new methods can achieve very high accuracy with very coarse mesh even for very small ε. In the contrast, the traditional finite element method does not get satisfactory numerical results with the same mesh.

Bloch decomposition-based Gaussian beam method for the Schrödinger equation with periodic potentials

The linear Schrodinger equation with periodic potentials is an important model in solid state physics. The most efficient direct simulation using a Bloch decomposition-based time-splitting spectral method [18] requires the mesh size to be O(@e) where @e is the scaled semiclassical parameter. In this paper, we generalize the Gaussian beam method introduced in Jin et al. [23] to solve this problem asymptotically. We combine the technique of Bloch decomposition and the Eulerian Gaussian beam method to arrive at an Eulerian computational method that requires mesh size of O(@e). The accuracy of this method is demonstrated via several numerical examples.

Tailored Finite Point Method for a Singular Perturbation Problem with Variable Coefficients in Two Dimensions

In this paper, we propose a tailored-finite-point method for a type of linear singular perturbation problem in two dimensions. Our finite point method has been tailored to some particular properties of the problem. Therefore, our new method can achieve very high accuracy with very coarse mesh even for very small ε, i.e. the boundary layers and interior layers do not need to be resolved numerically. In our numerical implementation, we study the classification of all the singular points for the corresponding degenerate first order linear dynamic system. We also study some cases with nonlinear coefficients. Our tailored finite point method is very efficient in both linear and nonlinear coefficients cases.

A Bloch Decomposition-Based Split-Step Pseudospectral Method for Quantum Dynamics with Periodic Potentials

We present a new numerical method for accurate computations of solutions to (linear) one-dimensional Schro¨dinger equations with periodic potentials. This is a prominent model in solid state physics where we also allow for perturbations by nonperiodic potentials describing external electric fields. Our approach is based on the classical Bloch decomposition method, which allows us to diagonalize the periodic part of the Hamiltonian operator. Hence, the dominant effects from dispersion and periodic lattice potential are computed together, while the nonperiodic potential acts only as a perturbation. Because the split-step communicator error between the periodic and nonperiodic parts is relatively small, the step size can be chosen substantially larger than for the traditional splitting of the dispersion and potential operators. Indeed it is shown by the given examples that our method is unconditionally stable and more efficient than the traditional split-step pseudospectral schemes. To this end a particular focus is on the semiclassical regime, where the new algorithm naturally incorporates the adiabatic splitting of slow and fast degrees of freedom.

Tailored finite point method for the interface problem

In this paper, we propose a tailored-finite-point method for a nu- merical simulation of the second order elliptic equation with discontinuous coefficients. Our finite point method has been tailored to someparticular properties of the problem, then we can get the approximate solution with the same behaviors as that of the exact solution very naturally. Especially, in one- dimensional case, when the coefficients are piecewise linearfunctions, we can get the exact solution with only one point in each subdomain. Furthermore, the stability analysis and the uniform convergence analysis in the energy norm are proved. On the other hand, our computational complexity is only O(N) for N discrete points. We also extend our method to two-dimensional problems.