On the Conclusion and Decision of Maxwell's Equations for the Problems with Given Wavefront
Keldysh Institute of Applied Mathematics, Russian Academy of Science.
The conclusion of the 4-D Maxwell's equations in coordinate system including self-time of an electromagnetic wave front is represented. The equations are determined; the correctness of variables replacement in the 3-equations is shown. The positive definiteness of electromagnetic field’s energy density is shown, the uniqueness of the decision of Gursa problem for the Maxwell's equations is proved. The locally one-dimensional finite-difference scheme for three-dimensional Maxwell’s equations is represented. The scheme destines for numerical solving of problems with initial data on the characteristic surface and has the second order of summary approximation in grid norm on uniform grids. Energy change theorem for presented scheme is constructed as a algebraic corollary of its equations. This theorem guarantees convergence of difference solution to exact solution with the second order in the energy norm. The convergence speed is checked by the comparison with analytical solutions.
Mathematical modelling in actual problems of science and technics