Arbitrary High Order Splitting Methods for Linear Schrödinger Equations with Non-trivial Compatibility Conditions

Jul 16, 2026

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Splitting methods provide a natural approach for numerical time integration of Schrödinger partial differential equations. While arbitrary high order splitting schemes exist for periodic boundary conditions, non-periodic settings typically trigger order reduction even from smooth initial data. The mechanism is traced to compatibility conditions that are not preserved by classical splitting. The work proposes modified splitting methods for one-dimensional linear Schrödinger equations with homogeneous Dirichlet boundaries, achieving arbitrary high order without order reduction, illustrated via a fourth-order scheme.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages30

Reading Time1 h 16 min

Introduction

Problem setup and motivation

Operator splitting and classical schemes

Order conditions and order reduction

Numerical illustration

arXiv :2607 .04835v1 [math .NA] 6 Jul 2026 ARBITRARY HIGH ORDER SPLITTING METHODS FOR LINEAR SCHRÖDINGER EQUATIONS WITH NON-TRIVIAL COMPATIBILITY CONDITIONS JOACKIM BERNIER 1 , RAMONA HÄBERLI2 , AND GILLES VILMART2 Abstract . Splitting methods are a natural choice for the numerical time integration of partial differential equations, and arbitrary high order splitting schemes exist for Schrödinger equations with periodic boundary conditions. However, in the presence of non-periodic boundary conditions, we show that they suffer in general from an order reduction, even for smooth initial conditions. The reason for such order reduction phenomena are so-called compatibility conditions, which are not preserved by classical splitting schemes. In this paper, we introduce a family of modified splitting methods for one-dimensional linear Schrödinger equations with homogeneous Dirichlet boundary conditions, which achieve an arbitrary high order, and do not suffer from any order reduction. This is illustrated with a fourth order splitting scheme considering initial conditions with various regularity properties. 1. Introduction We aim at studying smooth solutions to the linear Schrödinger equation (1 . 1) i∂tu = (∂2x + V )u in R × (0 , 1), u(·, 0) = u( · , 1) = 0 in R, where V ∈ C∞ ([0, 1];R) is a given potential and u : R × [0 , 1] → C. More precisely, we look for solutions (1 .2) u ∈ C0 (R;H2k ) ∩ C1 (R;H2k−2) for some k ≥ 1, where the Sobolev spaces are defined as usual by Hk = {u ∈ L2 ([0 , 1];C) | ∀ℓ ≤ k, ∂ℓxu ∈ L2 ([0 , 1];C)} . In general, Schrödinger equations serve as the fundamental evolution model for driven quantum systems, e.g. in optics [31], quantum fluids [7 , 8], and laser-matter interaction such as strong-field ionization [33] . The one-dimensional linear problem (1.1) is of use to linearize nonlinear problems and approximates higher-dimensional confined quantum systems, especially in the context of splitting methods. Thereby, the problem is often studied over a torus, i.e. with periodic boundary conditions [13 , 18 , 28], see also [22 , 23], where low regular initial data is considered. In contrast, on an unbounded domain, usually one inserts absorbing boundary conditions for the time integration of the problem [2 , 3 , 4 , 5] . However, when modeling a particle confined to a bounded region, we impose homogeneous Dirichlet boundary conditions. A common way to integrate in time the boundary value problem (1.1) on a time inverval [0, T] for some T > 0, is to approximate the exact flow by a numerical scheme at time tn = nτ, where 1 Nantes Université, CNRS, Laboratoire de Mathématiques Jean Leray, LMJL, F-44000 Nantes, France 2 Université de Genève, Section de mathématiques, 7-9 rue du Conseil-Général, CH-1211 Genève 4, Schwitzerland E-mail addresses: [joackim.bernier@univ-nantes.fr](joackim.bernier@univ-nantes.fr) , [ramona.haeberli@unige.ch](ramona.haeberli@unige.ch) , [gilles.vilmart@unige.ch](gilles.vilmart@unige.ch). 2 J. BERNIER, R. HÄBERLI, AND G. VILMART τ > 0 is the time step. We write un+1 = Φτun ≈ u (tn+1) = e−iτ(∂2x+V )u (tn), n = 0 , 1 , where Φτ is one step of a numerical scheme, e.g. a splitting method, which approximates the exact flow e−iτ(∂2x+V ) . We consider two subproblems, namely (1 .3) i∂tu(t, x) = ∂2xu (t, x) in R × (0 , 1), u(t,0) = u(t, 1) = 0 in R, as well as the potential equation (1.4) i∂tu(t, x) = V (x)u (t, x) in R × (0 , 1) , where we denote the exact flows by e−it∂2xu(0) and e−itVu(0) for given initial data u(0, ·) = u(0) and t ∈ (0, T] . Although an operator splitting is not absolutely necessary for the time integration of the linear problem (1.1), splitting schemes allow a separate treatment of each sub-operator, which can lead to increased efficiency and easier implementation. In [21] (see also [6] for lower regularity assumptions), second order convergence for the Strang splitting scheme (1 .5) ΦStrangτ = e − ~~iτ~~2 Ve−iτ∂2xe − ~~iτ

FAQs

Why do classical high order splitting methods experience order reduction for non-periodic boundary conditions?

They suffer order reduction because compatibility conditions are introduced by non-periodic boundaries and are not preserved by classical splitting schemes.

Which boundary conditions and equation type are targeted by the proposed modified methods?

The paper focuses on one-dimensional linear Schrödinger equations with homogeneous Dirichlet boundary conditions.

How does the paper demonstrate that order reduction is avoided?

It introduces a family of modified splitting methods that achieve arbitrary high order without order reduction, illustrated through a fourth-order splitting scheme under initial data with varying regularity.

Jul 16, 2026

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