Finite difference method

Abstract: The electromagnetic fields surrounding a thin, subseabed resistive disk in response to a deep-towed, time-harmonic electric dipole antenna are investigated using a newly developed 3D Cartesian, staggered-grid modeling algorithm. We demonstrate that finite-difference and finite-volume methods for solving the governing curl-curl equation yield identical, complex-symmetric coefficient matrices for the resulting N×N linear system of equations. However, the finite-volume approach has an advantage in that it naturally admits quadrature integration methods for accurate representation of highly compact or exponentially varying source terms constituting the right side of the resulting linear system of equations. This linear system is solved using a coupled two-term recurrence, quasi-minimal residual algorithm that doesnot require explicit storage of the coefficient matrix, thus reducing storage costs from 22N to 10N complex, double-precision words with no decrease in computational performance. The disk model serve...

Abstract: In this paper, the Alternating Group Explicit (AGE) method is developed and applied to derive the solution of a 2 point boundary value problem. The analysis clearly shows the method to be analogous to the A.D.I. method. The extension of the method to ultidimensional problems and techniques for improving the convergence rate and attaining higher order accuracy are also given.

Abstract: A fast algorithm is presented for solving electromagnetic scattering from a rectangular open cavity embedded in an infinite ground plane The medium inside the cavity is assumed to be (vertically) layered By introducing a transparent (artificial) boundary condition, the problem in the open cavity is reduced to a bounded domain problem A simple finite difference method is then applied to solve the model Helmholtz equation The fast algorithm is designed for solving the resulting discrete system in terms of the discrete Fourier transform in the horizontal direction, a Gaussian elimination in the vertical direction, and a preconditioning conjugate gradient method with a complex diagonal preconditioner for the indefinite interface system The existence and uniqueness of the finite difference solution are established for arbitrary wave numbers Our numerical experiments for large numbers of mesh points, up to 16 million unknowns, and for large wave numbers, eg, between 100 and 200 wavelengths, show that the algorithm is extremely efficient The cost for calculating the radar cross section, which is of significant interest in practice, is O(M2) for an $M \times M$ mesh The proposed algorithm may be extended easily to solve discrete systems from other discretization methods of the model problem

Abstract: A comprehensive study is presented regarding the numerical stability of the simple and common forward Euler explicit integration technique combined with some common finite difference spatial discretizations applied to the advection-diffusion equation. One-dimensional results are obtained using both the matrix method (for several boundary conditions) and the classical von Neumann method of stability analysis and arguments presented showing that the latter is generally to be preferred, regardless of the type of boundary conditions. The less-well-known Godunov-Ryabenkii theory is also applied for a particular (Robin) boundary condition. After verifying portions of the one-dimensional theory with some numerical results, the stabilities of the two- and three-dimensional equations are addressed using the von Neumann method and results presented in the form of a new stability theorem. Extension of a useful scheme from one dimension, where the pure advection limit is known variously as Leith's method or a Lax-Wendroff method, to many dimensions via finite elements is also addressed and some stability results presented.