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<p>The purpose of this paper is to derive a Crank-Nicolson finite difference scheme for the Klein-Gordon-Schrödinger (KGS) equations with nonlinear loss. Under appropriate regularity assumptions, we establish the existence and uniqueness of the fully discrete solution by analyzing the boundedness of discrete mass and energy. Furthermore, we derive a priori error estimates in the discrete <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(H_0^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>H</mi> <mn>0</mn> <mn>1</mn> </msubsup> </math></EquationSource> </InlineEquation> norms, showing that the scheme attains second-order convergence in both time and space. Numerical experiments are conducted to validate the theoretical analysis and demonstrate the accuracy and stability of the proposed scheme.</p>

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Error Estimates of Crank-Nicolson Finite Difference Scheme for Klein-Gordon-Schrödinger Equations with Nonlinear Damping

  • 期刊论文

  • 发表日期：

    2026年2月23日

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  • Anh Ha Le

摘要

The purpose of this paper is to derive a Crank-Nicolson finite difference scheme for the Klein-Gordon-Schrödinger (KGS) equations with nonlinear loss. Under appropriate regularity assumptions, we establish the existence and uniqueness of the fully discrete solution by analyzing the boundedness of discrete mass and energy. Furthermore, we derive a priori error estimates in the discrete \(L^2\) L 2 and \(H_0^1\) H 0 1 norms, showing that the scheme attains second-order convergence in both time and space. Numerical experiments are conducted to validate the theoretical analysis and demonstrate the accuracy and stability of the proposed scheme.