Linearly localized difference schemes for the nonlinear Maxwell model of a magnetic field into a substance

A linearly localized difference scheme with the first-order time approximation, is proposed for solving a nonlinear Maxwell model associated with the penetration of a magnetic field into a substance. The new scheme is computationally efficient since the resulting algebra equations are linear and can be computed by the fast Thomas algorithm without any Newton-type inner iterations. It is also local in time, that is, only numerical solutions in one previous time-level are necessary to update the current solutions, such that it requires much less storage compared with the fully implicit method. Furthermore, the exponential decaying behavior of difference solution, which is analogous to that of the continuous solution, is obtained. To improve the time accuracy, we apply the Crank–Nicolson-type time discretization to construct a second-order linearly localized method. Numerical examples are presented to support our theoretical results.