Luming Zhang

A finite difference scheme for generalized regularized long-wave equation

In this paper, a finite difference method for a Cauchy problem of generalized regularized long-wave (GRLW) equation was considered. An energy conservative finite difference scheme was proposed. Convergence and stability of the difference solution were proved. Numerical results demonstrate that the method is efficient and reliable.

Convergence of a conservative difference scheme for a class of Klein-Gordon-Schrödinger equations in one space dimension

A conservative difference scheme is presented for the initial-boundary value problem of a class of Klein-Gordon-Schrodinger equations. The scheme can be implicit or implicit-explicit, depending on the choice of a parameter. On the basis of the priori estimates and an inequality about norms, convergence of the difference solution is proved with order O(h^2+@t^2) in the energy norm.

New conservative difference schemes for a coupled nonlinear Schrödinger system

Abstract In this paper, two conservative difference schemes for solving a coupled nonlinear Schrodinger (CNLS) system are numerically analyzed. Firstly, a nonlinear implicit two-level finite difference scheme for CNLS system is studied, then a linear three-level difference scheme for CNLS system is presented. An induction argument and the discrete energy method are used to prove the second-order convergence and unconditional stability of the linear scheme. Numerical examples show the efficiency of the new scheme.

A conservative numerical scheme for a class of nonlinear Schrödinger equation with wave operator

In this paper, the initial-boundary value problem of a class of nonlinear Schrodinger equation with wave operator is considered. An explicit and efficient finite difference scheme is presented. This is a scheme of four levels with a discrete conservative law. Convergence and stability are proved.

On finite difference methods for fourth-order fractional diffusion–wave and subdiffusion systems ☆

Abstract In this paper, firstly, the finite difference method is explored for the fourth-order fractional diffusion–wave system. The method is proved to be uniquely solvable, stable and convergent in l ∞ -norm by the energy method. Then we examine a subdiffusion system and present the numerical analysis using a different method. Numerical experiments are provided to demonstrate the accuracy and efficiency of the proposed schemes.

Conservative schemes for the symmetric regularized Long Wave equations

In this paper, we study the Symmetric Regularized Long Wave (SRLW) equations by finite difference method. We design some numerical schemes which preserve the original conservative properties for the equations. The first scheme is two-level and nonlinear-implicit. Existence of its difference solutions are proved by Brouwer fixed point theorem. It is proved by the discrete energy method that the scheme is uniquely solvable, unconditionally stable and second-order convergent for U in L∞ norm, and for N in L2 norm on the basis of the priori estimates. The second scheme is three-level and linear-implicit. Its stability and second-order convergence are proved. Both of the two schemes are conservative so can be used for long time computation. However, they are coupled in computing so need more CPU time. Thus we propose another three-level linear scheme which is not only conservative but also uncoupled in computation, and give the numerical analysis on it. Numerical experiments demonstrate that the schemes are accurate and efficient.

A compact finite difference scheme for the nonlinear Schrödinger equation with wave operator

Abstract In this paper, a compact finite difference scheme is presented for an periodic initial value problem of the nonlinear Schrodinger (NLS) equation with wave operator. This is a scheme of three levels with a discrete conservation law. The unconditional stability and convergence in maximum norm with order O ( h 4 + τ 2 ) are proved by the energy method. A numerical experiment is presented to support our theoretical results.

Split-step orthogonal spline collocation methods for nonlinear Schrödinger equations in one, two, and three dimensions

Abstract Split-step orthogonal spline collocation (OSC) methods are proposed for one-, two-, and three-dimensional nonlinear Schrodinger (NLS) equations with time-dependent potentials. Firstly, the NLS equation is split into two nonlinear equations, and one or more one-dimensional linear equations. Commonly, the nonlinear subproblems could be integrated directly and accurately, but it fails when the time-dependent potential cannot be integrated exactly. In this case, we propose three approximations by using quadrature formulae, but the split order is not reduced. Discrete-time OSC schemes are applied for the linear subproblems. In numerical experiments, many tests are carried out to prove the reliability and efficiency of the split-step OSC (SSOSC) methods. Solitons in one, two, and three dimensions are well simulated, and conservative properties and convergence rates are demonstrated. We also apply the ways of solving the nonlinear subproblems to the split-step finite difference (SSFD) methods and the time-splitting spectral (TSSP) methods, and the approximate ways still work well. Finally, we apply the SSOSC methods to solve some problems of Bose–Einstein condensates.

Analysis of a symplectic difference scheme for a coupled nonlinear Schrödinger system

In general, proofs of convergence and stability are difficult for symplectic schemes of nonlinear equations. In this paper, a symplectic difference scheme is proposed for an initial-boundary value problem of a coupled nonlinear Schrodinger system. An important lemma and an induction argument are used to prove the unique solvability, convergence and stability of numerical solutions. An iterative algorithm is also proposed for the symplectic scheme and its convergence is proved. Numerical examples show the efficiency of the symplectic scheme and the correction of our numerical analysis.

A class of conservative orthogonal spline collocation schemes for solving coupled Klein–Gordon–Schrödinger equations

Abstract A class of discrete-time orthogonal spline collocation schemes for solving coupled Klein–Gordon–Schrodinger equations with initial and boundary conditions are considered. These schemes are constructed by using piecewise cubic Hermite interpolations in space combined with finite difference methods in time. It is proved that the schemes have the conservation laws of discrete energy, and possess second order accuracy in maximum norm and fourth order in L 2 -norm for time and space, respectively. The conservation laws and rate of convergence are verified in numerical experiments. Moreover, the propagations of solitary waves and collisions of two head-on solitary waves are also well simulated.