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<p>In this paper, we propose a novel space-time Legendre-Gauss-Lobatto collocation method for solving the time-dependent two-dimensional Schrödinger equation with nonhomogeneous boundary conditions. We first develop a new approach for systems of ordinary differential equations in the complex domain, utilizing the multi-domain Legendre-Gauss-Lobatto collocation method. We then derive the spectral rate of convergence for the proposed method in the discrete <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11075_2025_2201_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-norm for the semi-discrete formulation. Numerical results demonstrate that our formulation achieves exponential convergence in both space and time, thereby validating the theoretical findings.</p>

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Space-time Legendre-Gauss-Lobatto collocation method for the two-dimensional Schrödinger equation

  • 期刊论文

  • 发表日期：

    2025年8月14日

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  • Yingying Shan,
  • Wenjie Liu

摘要

In this paper, we propose a novel space-time Legendre-Gauss-Lobatto collocation method for solving the time-dependent two-dimensional Schrödinger equation with nonhomogeneous boundary conditions. We first develop a new approach for systems of ordinary differential equations in the complex domain, utilizing the multi-domain Legendre-Gauss-Lobatto collocation method. We then derive the spectral rate of convergence for the proposed method in the discrete \(L^2\) L 2 -norm for the semi-discrete formulation. Numerical results demonstrate that our formulation achieves exponential convergence in both space and time, thereby validating the theoretical findings.