Finite difference method

Abstract: The auxiliary differential equation method used for treating Lorentz media with the FDTD algorithm is reinterpreted to reduce computer memory requirements while maintaining full time synchronism. This new formulation enables us to save up to 20% of computer memory when compared to the previous approach. The validity of our formulation is proved analytically by comparing the resulting equations with those from the previous method. We also employ this approach for Debye media and show that no disadvantage arises in terms of memory requirement.

Abstract: A non-standard finite difference scheme is developed to solve the linear partial differential equations with time- and space-fractional derivatives. The Grunwald-Letnikov method is used to approximate the fractional derivatives. Numerical illustrations that include the linear inhomogeneous time-fractional equation, linear space-fractional telegraph equation, linear inhomogeneous fractional Burgers equation and the fractional wave equation are investigated to show the pertinent features of the technique. Numerical results are presented graphically and reveal that the non-standard finite difference scheme is very effective and convenient for solving linear partial differential equations of fractional order.

Abstract: The study of new waveguide shapes requires an accurate knowledge of their higher order modes for bandwidth consideration, waveguide discontinuity analysis, and multimode launching and propagation studies. The finite-difference method solved by successive overrelaxation is a very accurate and general technique for dominant mode solution. The method makes minimum demand on computer store, the number of storage locations required being equal to the order of the matrix defining the system of linear equations used. Convergence criteria require that this matrix be positive semidefinite. For modes higher than the dominant, the method fails to converge as this condition is violated. Solution is obtained by redefining the problem such that the matrix is positive semidefinite for all modes. An algorithm is described which produces one row at a time of the new matrix, as required for successive overrelaxation, and thus reserves almost all computer store for the eigenvector. The eigenvector elements give field potentials at discrete points in the guide cross section. Iteration for higher order modes is successfully applied with all the computational benefits realized by previous finite-difference schemes for the dominant mode. Examples studied are the rectangular, circular, asymmetric ridge, and lunar guides. Results for higher order cutoff wave numbers are usually accurate to within 0.1 percent.

Abstract: Convergence proofs for the multi-grid iteration are known for the case of finite element equations and for the case of some difference schemes discretizing boundary value problems in a rectangular region. In the present paper we give criteria of convergence that apply to general difference schemes for boundary value problems in Lipschitzian regions. Furthermore, convergence is proved for the multi-grid algorithm with Gauss-Seidel's iteration as smoothing procedure.