Related papers: Adaptive splitting methods for nonlinear Schr\"{o}…
This article is devoted to the construction of numerical methods which remain insensitive to the smallness of the semiclassical parameter for the linear Schr{\"o}dinger equation in the semiclassical limit. We specifically analyse the…
We consider Lie and Strang splitting for the time integration of constrained partial differential equations with a nonlinear reaction term. Since such systems are known to be sensitive with respect to perturbations, the splitting procedure…
We prove an error estimate for a Lie-Trotter splitting operator associated to the Schrodinger-Poisson equation in the semiclassical regime, when the WKB approximation is valid. In finite time, and so long as the solution to a compressible…
This paper introduces an adaptive time splitting technique for the solution of stiff evolutionary PDEs that guarantees an effective error control of the simulation, independent of the fastest physical time scale for highly unsteady…
Time dependent Schr\"odinger equations with conservative force field U commonly constitute a major challenge in the numerical approximation, especially when they are analysed in the semiclassical regime. Extremely high oscillations…
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 study the Strang splitting scheme for quasilinear Schr\"odinger equations. We establish the convergence of the scheme for solutions with small initial data. We analyze the linear instability of the numerical scheme, which explains the…
Splitting methods constitute a widely used class of numerical integrators for ordinary and partial differential equations, particularly well suited to problems that can be decomposed into simpler subproblems. High-order splitting schemes…
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 study the effective approximation for a nonlocal stochastic Schrodinger equation with a rapidly oscillating, periodically time-dependent potential. We use the natural diffusive scaling of heterogeneous system and study the limit…
In this paper we propose a modified Lie-type spectral splitting approximation where the external potential is of quadratic type. It is proved that we can approximate the solution to a one-dimensional nonlinear Schroedinger equation by…
In this paper we consider splitting methods for the time integration of parabolic and certain classes of hyperbolic partial differential equations, where one partial flow can not be computed exactly. Instead, we use a numerical…
We provide general product formulas for the solutions of non-autonomous abstract Cauchy problems. The main technical tool is the application of evolution semigroup methods, allowing the direct application of existing results on autonomous…
In this paper we describe an iterative operator-splitting method for unbounded operators. We derive error bounds for iterative splitting methods in the presence of unbounded operators and semigroup operators. Here mixed applications of…
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…
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…
We assess the applicability and efficiency of time-adaptive high-order splitting methods applied for the numerical solution of (systems of) nonlinear parabolic problems under periodic boundary conditions. We discuss in particular several…
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,…
Prior work on computable defect-based local error estimators for (linear) time-reversible integrators is extended to nonlinear and nonautonomous evolution equations. We prove that the asymptotic results from the linear case [W. Auzinger and…
We establish error bounds of the Lie-Trotter time-splitting sine pseudospectral method for the nonlinear Schr\"odinger equation (NLSE) with semi-smooth nonlinearity $ f(\rho) = \rho^\sigma$, where $\rho=|\psi|^2$ is the density with $\psi$…