Low regularity exponential-type integrators for semilinear Schrödinger equations

DOI10.1007/s10208-017-9352-1zbMath1402.65098arXiv1603.07746OpenAlexW2618966431WikidataQ59528935 ScholiaQ59528935MaRDI QIDQ1656378

Katharina Schratz, Alexander Ostermann

Publication date: 10 August 2018

Published in: Foundations of Computational Mathematics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1603.07746

zbMATH Keywords

convergencenonlinear Schrödinger equationslow regularityexponential-type time integrator

Mathematics Subject Classification ID

Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs (65M12) NLS equations (nonlinear Schrödinger equations) (35Q55) Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs (65M70) Time-dependent Schrödinger equations and Dirac equations (35Q41)

Related Items (45)

Analysis of fully discrete finite element methods for 2D Navier–Stokes equations with critical initial data ⋮ Uniformly Accurate Nested Picard Iterative Integrators for the Nonlinear Dirac Equation in the Nonrelativistic Regime ⋮ A first-order Fourier integrator for the nonlinear Schrödinger equation on 𝕋 without loss of regularity ⋮ Uniformly Accurate Low Regularity Integrators for the Klein--Gordon Equation from the Classical to NonRelativistic Limit Regime ⋮ Low-regularity integrator for the Davey-Stewartson system: elliptic-elliptic case ⋮ Uniformly accurate nested Picard iterative integrators for the Klein-Gordon equation in the nonrelativistic regime ⋮ Embedded exponential-type low-regularity integrators for KdV equation under rough data ⋮ An Embedded Exponential-Type Low-Regularity Integrator for mKdV Equation ⋮ A note on optimal \(H^1\)-error estimates for Crank-Nicolson approximations to the nonlinear Schrödinger equation ⋮ Time integrators for dispersive equations in the long wave regime ⋮ A symmetric low-regularity integrator for nonlinear Klein-Gordon equation ⋮ A second-order low-regularity correction of Lie splitting for the semilinear Klein–Gordon equation ⋮ On numerical approaches to nonlinear Schrödinger and Korteweg-de Vries equations for piecewise smooth and slowly decaying initial data ⋮ On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime ⋮ Fourier integrator for periodic NLS: low regularity estimates via discrete Bourgain spaces ⋮ A Semi-implicit Exponential Low-Regularity Integrator for the Navier--Stokes Equations ⋮ Regularized Numerical Methods for the Nonlinear Schrödinger Equation with Singular Nonlinearity ⋮ Uniformly accurate nested Picard iterative schemes for nonlinear Schrödinger equation with highly oscillatory potential ⋮ Fully discrete error analysis of first‐order low regularity integrators for the Allen‐Cahn equation ⋮ A semi-discrete first-order low regularity exponential integrator for the ``good Boussinesq equation without loss of regularity ⋮ Low-regularity integrator for the Davey-Stewartson II system ⋮ Low regularity integrators for semilinear parabolic equations with maximum bound principles ⋮ Numerical Integration of Schrödinger Maps via the Hasimoto Transform ⋮ Gauge-Transformed Exponential Integrator for Generalized KdV Equations with Rough Data ⋮ Optimal Error Bounds on the Exponential Wave Integrator for the Nonlinear Schrödinger Equation with Low Regularity Potential and Nonlinearity ⋮ Low-regularity integrators for nonlinear Dirac equations ⋮ Uniformly Accurate Oscillatory Integrators for the Klein--Gordon--Zakharov System from Low- to High-Plasma Frequency Regimes ⋮ Comparison of numerical methods for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime ⋮ Numerical integrators for continuous disordered nonlinear Schrödinger equation ⋮ A fully discrete low-regularity integrator for the 1D periodic cubic nonlinear Schrödinger equation ⋮ Asymptotic consistent exponential-type integrators for Klein-Gordon-Schrödinger systems from relativistic to non-relativistic regimes ⋮ On the splitting method for the nonlinear Schrödinger equation with initial data in \(H^1\) ⋮ Unnamed Item ⋮ Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity ⋮ LONG TIME BEHAVIOR OF THE SOLUTIONS OF NLW ON THE -DIMENSIONAL TORUS ⋮ Uniformly Accurate Nested Picard Iterative Integrators for the Dirac Equation in the Nonrelativistic Limit Regime ⋮ A Fourier Integrator for the Cubic Nonlinear Schrödinger Equation with Rough Initial Data ⋮ A General Framework of Low Regularity Integrators ⋮ Error analysis of a class of semi-discrete schemes for solving the Gross-Pitaevskii equation at low regularity ⋮ Two exponential-type integrators for the ``good Boussinesq equation ⋮ On nested Picard iterative integrators for highly oscillatory second-order differential equations ⋮ Resonance-based schemes for dispersive equations via decorated trees ⋮ BAMPHI: matrix-free and transpose-free action of linear combinations of \(\varphi\)-functions from exponential integrators ⋮ Optimal convergence and long-time conservation of exponential integration for Schrödinger equations in a normal or highly oscillatory regime ⋮ A fully discrete low-regularity integrator for the nonlinear Schrödinger equation

Cites Work

• Unnamed Item
• Unnamed Item
• Fractional error estimates of splitting schemes for the nonlinear Schrödinger equation
• The quintic NLS as the mean field limit of a boson gas with three-body interactions
• A splitting method for the nonlinear Schrödinger equation
• Geometric numerical integration and Schrödinger equations
• One-stage exponential integrators for nonlinear Schrödinger equations over long times
• Sharp well-posedness and ill-posedness results for a quadratic nonlinear Schrödinger equation
• Low-regularity bilinear estimates for a quadratic nonlinear Schrödinger equation
• Symmetric exponential integrators with an application to the cubic Schrödinger equation
• Splitting methods for partial differential equations with rough solutions. Analysis and Matlab programs
• Exponential Runge-Kutta methods for the Schrödinger equation
• Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II: The KdV-equation
• On the ill-posedness of some canonical dispersive equations.
• Counterexamples to bilinear estimates related with the KdV equation and the nonlinear Schrödinger equation
• Exponential time integration of solitary waves of cubic Schrödinger equation
• An exponential-type integrator for the KdV equation
• Multilinear weighted convolution of L 2 functions, and applications to nonlinear dispersive equations
• Exponential integrators
• Convergence of a split-step Hermite method for the Gross-Pitaevskii equation
• On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations
• Solving Ordinary Differential Equations I
• Splitting methods
• The cauchy problem for the critical nonlinear Schrödinger equation in Hs
• Global Solutions for 3D Quadratic Schrodinger Equations
• Order Estimates in Time of Splitting Methods for the Nonlinear Schrödinger Equation
• Quadratic forms for the 1-D semilinear Schrödinger equation
• Convergence Analysis of High-Order Time-Splitting Pseudospectral Methods for Nonlinear Schrödinger Equations
• Fourth-Order Time-Stepping for Stiff PDEs
• Geometric Numerical Integration
• Generalized Runge-Kutta Processes for Stable Systems with Large Lipschitz Constants

This page was built for publication: Low regularity exponential-type integrators for semilinear Schrödinger equations