Chunxiong Zheng

Exact nonreflecting boundary conditions for one-dimensional cubic nonlinear Schrödinger equations

The numerical approximation of one-dimensional cubic nonlinear Schrodinger equations on the whole real axis is studied in this paper. Based on the work of A. Boutet de Monvel, A.S. Fokas and D. Shepelsky [Lett. Math. Phys., 65(3): 199-212, 2003], a kind of exact nonreflecting boundary conditions are derived on the artificially introduced boundary points. The related numerical issues are discussed in detail. Several numerical tests are performed to demonstrate the behaviour of the proposed scheme.

A perfectly matched layer approach to the nonlinear Schrödinger wave equations

Absorbing boundary conditions (ABCs) are generally required for simulating waves in unbounded domains. As one of those approaches for designing ABCs, perfectly matched layer (PML) has achieved great success for both linear and nonlinear wave equations. In this paper we apply PML to the nonlinear Schrodinger wave equations. The idea involved is stimulated by the good performance of PML for the linear Schrodinger equation with constant potentials, together with the time-transverse invariant property held by the nonlinear Schrodinger wave equations. Numerical tests demonstrate the effectiveness of our PML approach for both nonlinear Schrodinger equations and some Schrodinger-coupled systems in each spatial dimension.

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.

Numerical Solution to the Sine-Gordon Equation Defined on the Whole Real Axis

Numerical simulation of the solution to the sine-Gordon equation on the whole real axis is considered in this paper. Based on nonlinear spectral analysis, exact nonreflecting boundary conditions are derived at two artificially introduced boundary points. Then a numerical scheme of second order is proposed to approximate the solution. In the end, some numerical examples are provided to demonstrate the effectiveness of the proposed scheme.

Exact artificial boundary conditions for problems with periodic structures

Based on the work of Zheng on the artificial boundary condition for the Schrodinger equation with sinusoidal potentials at infinity, an analytical impedance expression is presented for general second-order ODE problems with periodic coefficients and its validity is shown to be strongly supported by numerical evidences. This new expression for the kernel of the Dirichlet-to-Neumann mapping of the artificial boundary conditions is then used for computing the bound states of the Schrodinger operator with periodic potentials at infinity. Other potential applications are associated with the exact artificial boundary conditions for some time-dependent problems with periodic structures. As an example, a two-dimensional hyperbolic equation modeling the TM polarization of the electromagnetic field with a periodic dielectric permittivity is considered.

Numerical scattering analysis of TE plane waves by a metallic diffraction grating with local defects

We consider the numerical scattering of plane waves by a metallic diffraction grating with a single defect. Besides different diffracted orders, a perturbed scattered field with arbitrary reflection direction is generated by the defect. We transform the diffraction grating into a closed waveguide by introducing a perfectly matched layer. The diffracted field is solved by applying pseudoperiodic boundary conditions on cell boundaries. Then we take two steps to resolve the perturbed scattered field. On the defect cell it is obtained by solving the governing wave equation with absorbing boundary conditions derived by a fast recursive doubling procedure. On the rest of the domain the perturbed scattered field is computed by using the recursive matrix operators efficiently. An optical theorem is employed to evaluate the proposed method.

Mixed finite element method and high-order local artificial boundary conditions for exterior problems of elliptic equation☆

The mixed finite element method is used to solve the exterior elliptic problem with high-order local artificial boundary conditions. New unknowns are introduced to reduce the order of the derivatives to two. This leads to an equivalent mixed variational problem such that the normal finite element can be used and special finite elements are no longer needed on the adjacent layer of the artificial boundary. Error estimates are obtained for some local artificial boundary conditions with prescribed order. Numerical examples are presented and the results demonstrate the effectiveness of this method.

Reconstruction of a penetrable obstacle in periodic waveguides

Abstract The reconstruction of a penetrable obstacle embedded in a periodic waveguide is a challenging problem. In this paper, the inverse problem is formulated as an optimization problem. We prove some properties of the scattering operator and propose an iterative scheme to approximate the support of the obstacle. Using the limiting absorption principle and a recursive doubling technique, we implement a fast algorithm based on a carefully designed finite element method for the forward scattering problem. Numerical examples validate the effectiveness of the method.