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

In this paper, we consider a nonlinear one-dimensional problem for the equations ofsuperconductivity theory. A specific feature of the problem is a nonstandard Robin type junctioncondition on the inner boundary and a discontinuous solution. An optimal homogeneousmonotone 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 completesystem of Usadel equations, which is the basic mathematical model at the microlevel fordescribing the currents and fields in superconductors with Josephson junctions. The results ofcalculations for the problem of a pellet with an Abrikosov vortex are presented.