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Abstract <p> The paper considers a one-dimensional problem for elliptic equations with nonstandardjump conditions on the internal boundary and a discontinuous solution. The integro-interpolation(balance) method is used to approximate the problem, including the junction condition on theinner boundary, which leads, in the case of Robin relations (the jump of the solution isproportional to the flux), to a four-point stencil. This finite-difference scheme is used to solve thesystem of nonlinear Usadel equations, which is the basic mathematical model at the microlevel fordescribing currents and fields in superconductors, including those with Josephson junctions. Theresults of calculations for the Abrikosov vortex problem are presented, and the accuracy of theproposed approach is investigated, in particular, for a simplified three-point scheme.</p>

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Finite-Difference Integro-Interpolation Method for Discontinuous Solutions of the Usadel Equations

  • 期刊论文

  • 发表日期：

    2025年7月1日

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  • M. M. Khapaev,
  • M. Yu. Kupriyanov

摘要

Abstract

The paper considers a one-dimensional problem for elliptic equations with nonstandardjump conditions on the internal boundary and a discontinuous solution. The integro-interpolation(balance) method is used to approximate the problem, including the junction condition on theinner boundary, which leads, in the case of Robin relations (the jump of the solution isproportional to the flux), to a four-point stencil. This finite-difference scheme is used to solve thesystem of nonlinear Usadel equations, which is the basic mathematical model at the microlevel fordescribing currents and fields in superconductors, including those with Josephson junctions. Theresults of calculations for the Abrikosov vortex problem are presented, and the accuracy of theproposed approach is investigated, in particular, for a simplified three-point scheme.