Tridiagonal matrix algorithm

Abstract: Qualitative properties of matrix splitting methods for linear systems with tridiagonal and block tridiagonal Stieltjes-Toeplitz matrices are studied. Two particular splittings, the so-called symmetric tridiagonal splittings and the bidiagonal splittings, are considered, and conditions for qualitative properties like nonnegativity and shape preservation are shown for them. Special attention is paid to their close relation to the well-known splitting techniques like regular and weak regular splitting methods. Extensions to block tridiagonal matrices are given, and their relation to algebraic representations of domain decomposition methods is discussed. The paper is concluded with illustrative numerical experiments.

Abstract: For MinRes and SymmLQ it is essential to compute a QR decomposition of a tridiagonal coefficient matrix gained in the Lanczos process. This QR decomposition is constructed by an update scheme applying in every step a single Givens rotation. Using complex Householder reflections we generalize this idea to block tridiagonal matrices that occur in generalizations of MinRes and SymmLQ to block methods for systems with multiple right-hand sides. Some implementation details are given, and we compare the method with an algorithm based on Givens rotations used in block QMR. Our approach generalizes to the QR decomposition of upper block Hessenberg matrices resulting from the block Arnoldi process and is applicable in block GMRes. Keywords— block Lanczos process, block Krylov space methods. I. The symmetric Lanczos algorithm In 1975 Christopher Paige and Michael Saunders [PAI 75] proposed two iterative Krylov subspace methods called MinRes and SymmLQ for solving sparse Hermitian indefinite linear systems

Abstract: : Dielectric Barrier Discharge (DBD) type devices, when used as plasma actuators, have shown significant promise for use in many aeronautical applications. Experimentally, DBD actuator devices have been shown to induce motion in initially still air, and to cause re-attachment of air flow over a wing surface at a high angle of attack. This thesis explores the numerical simulation of the DBD device in both a lD and 2D environment. Using well established fluid equation techniques, along with the appropriate approximations for the regime under which these devices will be operating, computational results for various conditions and geometries are explored. In order to validate the code, results are compared to analytic or experimental data whenever possible, or matched with other similar numeric simulations to help establish the accuracy of the code. Solutions to Poisson's equation for the potential, electron and ion continuity equations, and the electron energy equation are solved semi-implicitly in a sequential manner. Each of the governing equations is solved by casting them into a tridiagonal grid, and using the computationally efficient Thomas algorithm to solve lD regions in a single iteration. The Scharfetter-Gummel flux discretization method is used to add stability to the code when transitioning from a field to diffusion dominated region or vice versa. Estimates for the ionization and recombination rates and for the transport coefficients of the background gas are calculated as a function of the local average electron energy, and updated for every calculation point in the domain on the completion of the solution to the electron energy equation.