Exponential Operator Splitting Research Articles

Convergence Analysis of High-Order Time-Splitting Pseudospectral Methods for Nonlinear Schrödinger Equations

In this work, the issue of favorable numerical methods for the space and time discretization of low-dimensional nonlinear Schrodinger equations is addressed. The objective is to provide a stability and error analysis of high-accuracy discretizations that rely on spectral and splitting methods. As a model problem, the time-dependent Gross--Pitaevskii equation arising in the description of Bose--Einstein condensates is considered. For the space discretization pseudospectral methods collocated at the associated quadrature nodes are analyzed. For the time integration high-order exponential operator splitting methods are studied, where the decomposition of the function defining the partial differential equation is chosen in accordance with the underlying spectral method. The convergence analysis relies on a general framework of abstract nonlinear evolution equations and fractional power spaces defined by the principal linear part. Essential tools in the derivation of a temporal global error estimate are furthe...

An exact local error representation of exponential operator splitting methods for evolutionary problems and applications to linear Schrödinger equations in the semi-classical regime

In this paper, we are concerned with the derivation of a local error representation for exponential operator splitting methods when applied to evolutionary problems that involve critical parameters. Employing an abstract formulation of differential equations on function spaces, our framework includes Schrodinger equations in the semi-classical regime as well as parabolic initial-boundary value problems with high spatial gradients. We illustrate the general mechanism on the basis of the first-order Lie splitting and the second-order Strang splitting method. Further, we specify the local error representation for a fourth-order splitting scheme by Yoshida. From the given error estimate it is concluded that higher-order exponential operator splitting methods are favourable for the time-integration of linear Schrodinger equations in the semi-classical regime with critical parameter 0<e≪1, provided that the time stepsize h is sufficiently smaller than $\sqrt[p]{\varepsilon}$ , where p denotes the order of the splitting method.

High-order time-splitting Hermite and Fourier spectral methods

In this paper, we are concerned with the numerical solution of the time-dependent Gross–Pitaevskii Equation (GPE) involving a quasi-harmonic potential. Primarily, we consider discretisations that are based on spectral methods in space and higher-order exponential operator splitting methods in time. The resulting methods are favourable in view of accuracy and efficiency; moreover, geometric properties of the equation such as particle number and energy conservation are well captured. Regarding the spatial discretisation of the GPE, we consider two approaches. In the unbounded domain, we employ a spectral decomposition of the solution into Hermite basis functions; on the other hand, restricting the equation to a sufficiently large bounded domain, Fourier techniques are applicable. For the time integration of the GPE, we study various exponential operator splitting methods of convergence orders two, four, and six. Our main objective is to provide accuracy and efficiency comparisons of exponential operator splitting Fourier and Hermite pseudospectral methods for the time evolution of the GPE. Furthermore, we illustrate the effectiveness of higher-order time-splitting methods compared to standard integrators in a long-term integration.

High-Order Exponential Operator Splitting Methods for Time-Dependent Schrödinger Equations

In this paper, we deduce high-order error bounds for exponential operator splitting methods. The employed techniques are specific to linear differential equations of the form $u'(t) = A \, u(t) + B \, u(t)$, $t \geq 0$, involving an unbounded operator $A$. In particular, evolutionary Schrodinger equations with sufficiently regular initial values are included in the analysis.

Exponential operator splitting time integration for spectral methods

Pseudospectral spatial discretization by orthogonal polynomials and Strang splitting method for time integration are applied to second-order linear evolutionary PDEs. Before such a numerical integration can be used the original PDE is transformed into a suitable form. Trigonometric, Jacobi (and some of their special cases), generalized Laguerre and Hermite polynomials are considered. A double representation of a function (by coefficients of a polynomial expansion and by values at the nodes associated with a suitable quadrature formula) is used for numerical implementation so that it is possible to avoid calculations of matrix exponentials.

Higher-order operator splitting methods for deterministic parabolic equations

The Sheng-Suzuki theorem states that all exponential operator splitting methods of order greater than 2 must contain negative time integration. There have been claims in the literature that higher-order splitting methods for deterministic parabolic equations are unstable due to this fact. We show stability for a class of higher-order splitting methods for integrating deterministic parabolic equations. We note that problems with backwards time integration will still exist for stochastic integration methods for which information is lost and backward timesteps become ill-defined. Therefore, completely positive splitting methods, such as those developed by Chin, still have an important place. We present numerical results from first-, second-, third- and fourth-order methods showing that the error becomes increasingly small as the order increases.