Related papers: Applying splitting methods with complex coefficien…
We consider the linear and non linear cubic Schr\"odinger equations with periodic boundary conditions, and their approximations by splitting methods. We prove that for a dense set of arbitrary small time steps, there exists numerical…
The nonlinear Schr\"odinger and the Schr\"odinger-Newton equations model many phenomena in various fields. Here, we perform an extensive numerical comparison between splitting methods (often employed to numerically solve these equations)…
Numerical solving the Schr\"odinger equation with incommensurate potentials presents a great challenge since its solutions could be space-filling quasiperiodic structures without translational symmetry nor decay. In this paper, we propose…
A scheme stemming from the use of pseudospectral approximations to spatial derivatives followed by a time integrator based on trigonometric polynomials is proposed for the numerical solutions of the coupled nonlinear Klein--Gordon…
We introduce efficient and robust exponential-type integrators for Klein-Gordon equations which resolve the solution in the relativistic regime as well as in the highly-oscillatory non-relativistic regime without any step-size restriction,…
Schr\"odinger equations with time-dependent potentials are of central importance in quantum physics and theoretical chemistry, where they aid in the simulation and design of systems and processes at atomic scales. Numerical approximation of…
This article is devoted to the construction of new numerical methods for the semiclassical Schr\"odinger equation. A phase-amplitude reformulation of the equation is described where the Planck constant epsilon is not a singular parameter.…
We consider the numerical integration of non-autonomous separable parabolic equations using high order splitting methods with complex coefficients (methods with real coefficients of order greater than two necessarily have negative…
Operator splitting methods combined with finite element spatial discretizations are studied for time-dependent nonlinear Schr\"odinger equations. In particular, the Schr\"odinger-Poisson equation under homogeneous Dirichlet boundary…
We provide a comprehensive survey of splitting and composition methods for the numerical integration of ordinary differential equations (ODEs). Splitting methods constitute an appropriate choice when the vector field associated with the ODE…
This work proposes and analyzes an efficient numerical method for solving the nonlinear Schr\"odinger equation with quasiperiodic potential, where the projection method is applied in space to account for the quasiperiodic structure and the…
We present a class of exponential integrators to compute solutions of the stochastic Schr\"odinger equation arising from the modeling of open quantum systems. In order to be able to implement the methods within the same framework as the…
The present work provides a comprehensive study of symmetric-conjugate operator splitting methods in the context of linear parabolic problems and demonstrates their additional benefits compared to symmetric splitting methods. Relevant…
In this paper we present splitting methods which are based on iterative schemes and applied to stochastic nonlinear Schroedinger equation. We will design stochastic integrators which almost conserve the symplectic structure. The idea is…
Standard numerical integrators suffer from an order reduction when applied to nonlinear Schr\"{o}dinger equations with low-regularity initial data. For example, standard Strang splitting requires the boundedness of the solution in $H^{r+4}$…
In approximating solutions of nonstationary problems, various approaches are used to compute the solution at a new time level from a number of simpler (sub-)problems. Among these approaches are splitting methods. Standard splitting schemes…
The efficient evaluation of high-dimensional integrals is of importance in both theoretical and practical fields of science, such as data science, statistical physics, and machine learning. However, exact computation methods suffer from the…
Most numerical methods for time integration use real time steps. Complex time steps provide an additional degree of freedom, as we can select the magnitude of the step in both the real and imaginary directions. By time stepping along…
In [8], some exact splittings are proposed for inhomogeneous quadratic differential equations including, for example, transport equations, kinetic equations, and Schr{\"o}dinger type equations with a rotation term. In this work, these exact…
We consider the nonlinear Schr{\"o}dinger equation with a potential, also known as Gross-Pitaevskii equation. By introducing a suitable spectral localization, we prove low regularity error estimates for the time discretization corresponding…