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 Schrödinger 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.     

Maximum norm error bound of a linearized difference scheme for a coupled nonlinear Schrödinger equations

In this article, a linearized conservative difference scheme for a coupled nonlinear Schrödinger equations is studied. The discrete energy method and an useful technique are used to analyze the difference scheme. It is shown that the difference solution unconditionally converges to the exact solution with second order in the maximum norm. Numerical experiments are presented to support the theoretical results.     

A splitting method for the nonlinear Schrödinger equation

We introduce a splitting method for the semilinear Schrödinger equation and prove its convergence for those nonlinearities which can be handled by the classical well-posedness L²(Rd)-theory. More precisely, we prove that the scheme is of first order in the L²(Rd)-norm for H²(Rd)-initial data.     

Lobatto IIIA-IIIB discretization of the strongly coupled nonlinear Schrödinger equation

In this paper, we construct a second order semi-explicit multi-symplectic integrator for the strongly coupled nonlinear Schrödinger equation based on the two-stage Lobatto IIIA-IIIB partitioned Runge-Kutta method. Numerical results for different solitary wave solutions including elastic and inelastic collisions, fusion of two solitons and with periodic solutions confirm the excellent long time behavior of the multi-symplectic integrator by preserving global energy, momentum and mass.     

Eigenvalue problem for a coupled channel Schrödinger equation with application to the description of deformed nuclear systems

We discuss the accurate computation of the eigensolutions of systems of coupled channel Schrödinger equations as they appear in studies of real physical phenomena like fission, alpha decay and proton emission. A specific technique is used to compute the solution near the singularity in the origin, while on the rest of the interval the solution is propagated using a piecewise perturbation method. Such a piecewise perturbation method allows us to take large steps even for high energy-values. We consider systems with a deformed potential leading to an eigenvalue problem where the energies are given and the required eigenvalue is related to the adjustment of the potential, viz, the eigenvalue is the depth of the nuclear potential. A shooting technique is presented to determine this eigenvalue accurately.     

Defect-based local error estimators for splitting methods,with application to Schrödinger equations,Part I: The linear case

We introduce a defect correction principle for exponential operator splitting methods applied to time-dependent linear Schrödinger equations and construct a posteriori local error estimators for the Lie–Trotter and Strang splitting methods. Under natural commutator bounds on the involved operators we prove asymptotical correctness of the local error estimators, and along the way recover the known a priori convergence bounds. Numerical examples illustrate the theoretical local and global error estimates.     

Convergence of a finite element scheme for the two-dimensional time-dependent Schrödinger equation in a long strip

This paper addresses the finite element method for the two-dimensional time-dependent Schrödinger equation on an infinite strip by using artificial boundary conditions. We first reduce the original problem into an initial-boundary value problem in a bounded domain by introducing a transparent boundary condition, then fully discretize this reduced problem by applying the Crank-Nicolson scheme in time and a bilinear or quadratic finite element approximation in space. This scheme, by a rigorous analysis, has been proved to be unconditionally stable and convergent, and its convergence order has also been obtained. Finally, two numerical examples are given to verify the accuracy of the scheme.     

A new high accuracy locally one-dimensional scheme for the wave equation

In this paper, a new locally one-dimensional (LOD) scheme with error of O(Δt⁴+h⁴) for the two-dimensional wave equation is presented. The new scheme is four layer in time and three layer in space. One main advantage of the new method is that only tridiagonal systems of linear algebraic equations have to be solved at each time step. The stability and dispersion analysis of the new scheme are given. The computations of the initial and boundary conditions for the two intermediate time layers are explicitly constructed, which makes the scheme suitable for performing practical simulation in wave propagation modeling. Furthermore, a comparison of our new scheme and the traditional finite difference scheme is given, which shows the superiority of our new method.     

Nonlinear scheme with high accuracy for nonlinear coupled parabolic-hyperbolic system

A nonlinear finite difference scheme with high accuracy is studied for a class of two-dimensional nonlinear coupled parabolic-hyperbolic system. Rigorous theoretical analysis is made for the stability and convergence properties of the scheme, which shows it is unconditionally stable and convergent with second order rate for both spatial and temporal variables. In the argument of theoretical results, difficulties arising from the nonlinearity and coupling between parabolic and hyperbolic equations are overcome, by an ingenious use of the method of energy estimation and inductive hypothesis reasoning. The reasoning method here differs from those used for linear implicit schemes, and can be widely applied to the studies of stability and convergence for a variety of nonlinear schemes for nonlinear PDE problems. Numerical tests verify the results of the theoretical analysis. Particularly it is shown that the scheme is more accurate and faster than a previous two-level nonlinear scheme with first order temporal accuracy.     

A parametric symmetric linear four-step method for the efficient integration of the Schrödinger equation and related oscillatory problems

In this article, we develop an explicit symmetric linear phase-fitted four-step method with a free coefficient as parameter. The parameter is used for the optimization of the method in order to solve efficiently the Schrödinger equation and related oscillatory problems. We evaluate the local truncation error and the interval of periodicity as functions of the parameter. We reveal a direct relationship between the periodicity interval and the local truncation error. We also measure the efficiency of the new method for a wide range of possible values of the parameter and compare it to other well known methods from the literature. The analysis and the numerical results help us to determine the optimal values of the parameter, which render the new method highly efficient.     

Numerical analysis for a conservative difference scheme to solve the Schrödinger-Boussinesq equation

In this paper, we present a finite difference scheme for the solution of an initial-boundary value problem of the Schrödinger-Boussinesq equation. The scheme is fully implicit and conserves two invariable quantities of the system. We investigate the existence of the solution for the scheme, give computational process for the numerical solution and prove convergence of iteration method by which a nonlinear algebra system for unknown Vⁿ⁺¹ is solved. On the basis of a priori estimates for a numerical solution, the uniqueness, convergence and stability for the difference solution is discussed. Numerical experiments verify the accuracy of our method.     

A convergent scheme for a non local Hamilton Jacobi equation modelling dislocation dynamics

We study dislocation dynamics with a level set point of view. The model we present here looks at the zero level set of the solution of a non local Hamilton Jacobi equation, as a dislocation in a plane of a crystal. The front has a normal speed, depending on the solution itself. We prove existence and uniqueness for short time in the set of continuous viscosity solutions. We also present a first order finite difference scheme for the corresponding level set formulation of the model. The scheme is based on monotone numerical Hamiltonian, proposed by Osher and Sethian. The non local character of the problem makes it not monotone. We obtain an explicit convergence rate of the approximate solution to the viscosity solution. We finally provide numerical simulations.This work has been supported by funds from ACI JC 1041 “Mouvements d’interfaces avec termes non-locaux”, from ACI-JC 1025 “Dynamique des dislocations” and from ONERA, Office National d’Etudes et de Recherches. The second author was also supported by the ENPC-Région Ile de France.     

A qualocation method for Burgers’ equation

In this paper, a qualocation method for the one-dimensional Burgers’ equation is proposed. A semidiscrete scheme along with optimal error estimates is discussed. Results of a numerical experiment performed support the theoretical results.     

Simple geometry facilitates iterative solution of a nonlinear equation via a special transformation to accelerate convergence to third order

Direct substitution _xk+1=g(_xk)$x_{k+1}=g(x_k)$ generally represents iterative techniques for locating a root z   of a nonlinear equation f(x)$f(x)$. At the solution, f(z)=0$f(z)=0$ and g(z)=z$g(z)=z$. Efforts continue worldwide both to improve old iterators and create new ones. This is a study of convergence acceleration by generating secondary solvers through the transformation _gm(x)=(g(x)-m(x)x)/(1-m(x))$g_m(x)=(g(x)-m(x)x)/(1-m(x))$ or, equivalently, through partial substitution _gmps(x)=x+G(x)(g-x)$g_{m\mathrm{ps}}(x)=x+G(x)(g-x)$, G(x)=1/(1-m(x))$G(x)=1/(1-m(x))$. As a matter of fact, _gm(x)≡_gmps(x)$g_m(x)\equiv g_{m\mathrm{ps}}(x)$ is the point of intersection of a linearised g   with the g=x$g=x$ line. Aitken's and Wegstein's accelerators are special cases of _gm$g_m$. Simple geometry suggests that m(x)=(g^′(x)+g^′(z))/2$m(x)=(g^{\prime }(x)+g^{\prime }(z))/2$ is a good approximation for the ideal slope of the linearised g  . Indeed, this renders a third-order _gm$g_m$. The pertinent asymptotic error constant has been determined. The theoretical background covers a critical review of several partial substitution variants of the well-known Newton's method, including third-order Halley's and Chebyshev's solvers. The new technique is illustrated using first-, second-, and third-order primaries. A flexible algorithm is added to facilitate applications to any solver. The transformed Newton's method is identical to Halley's. The use of m(x)=(g^′(x)+g^′(z))/2$m(x)=(g^{\prime }(x)+g^{\prime }(z))/2$ thus obviates the requirement for the second derivative of f(x)$f(x)$. Comparison and combination with Halley's and Chebyshev's solvers are provided. Numerical results are from the square root and cube root examples.     

The exponential integrator scheme for stochastic partial differential equations: Pathwise error bounds

We present an error analysis for the pathwise approximation of a general semilinear stochastic evolution equation in d dimensions. We discretise in space by a Galerkin method and in time by using a stochastic exponential integrator. We show that for spatially regular (smooth) noise the number of nodes needed for the noise can be reduced and that the rate of convergence degrades as the regularity of the noise reduces (and the noise becomes rougher).     

High order compact Alternating Direction Implicit method for the generalized sine-Gordon equation

High order compact Alternating Direction Implicit scheme is given for solving the generalized sine-Gordon equation in a two-dimensional rectangular domain. We apply the compact finite difference operators to obtain a fourth order discretization for the second order space derivatives, and we give a linearized three time level algorithm for solving the original nonlinear equation. Error estimate is given by the energy method. Numerical results are provided to verify the accuracy and efficiency of this algorithm.     

A high order uniformly convergent alternating direction scheme for time dependent reaction–diffusion singularly perturbed problems

In this work we design and analyze an efficient numerical method to solve two dimensional initial-boundary value reaction–diffusion problems, for which the diffusion parameter can be very small with respect to the reaction term. The method is defined by combining the Peaceman and Rachford alternating direction method to discretize in time, together with a HODIE finite difference scheme constructed on a tailored mesh. We prove that the resulting scheme is ε-uniformly convergent of second order in time and of third order in spatial variables. Some numerical examples illustrate the efficiency of the method and the orders of uniform convergence proved theoretically. We also show that it is easy to avoid the well-known order reduction phenomenon, which is usually produced in the time integration process when the boundary conditions are time dependent. This research has been partially supported by the project MEC/FEDER MTM2004-01905 and the Diputación General de Aragón.     

The role of coefficients of a general SPDE on the stability and convergence of a finite difference method

In this paper for the approximate solution of stochastic partial differential equations (SPDEs) of Itô-type, the stability and application of a class of finite difference method with regard to the coefficients in the equations is analyzed. The finite difference methods discussed here will be either explicit or implicit and a comparison between them will be reported. We prove the consistency and stability of these methods and investigate the influence of the multiplier (particularly multiplier of the random noise) in mean square stability. From stochastic version of Lax-Richtmyer the convergence of these methods under some conditions are established. Numerical experiments are included to show the efficiency of the methods.