A Finite-Difference Scheme for Discontinuous Solutions of the Usadel Equations

In this paper, we consider a nonlinear one-dimensional problem for the equations of superconductivity theory. A specific feature of the problem is a nonstandard Robin type junction condition on the inner boundary and a discontinuous solution. An optimal homogeneous monotone difference scheme including a condition on the interface is constructed for the problem. By solving a series of elliptic problems and by using Newton’s method, we solve the complete system of Usadel equations, which is the basic mathematical model at the microlevel for describing the currents and fields in superconductors with Josephson junctions. The results of calculations for the problem of a pellet with an Abrikosov vortex are presented.