The (Extrapolated) Alternating Direction Implicit Preconditioners for the class of Conjugate Gradient Methods are applied for the solution of the second order elliptic equation in a rectangle under Dirichlet boundary conditions. The PDE is approximated by uniform meshes of 5− and 9−point difference schemes and analytic expressions for the optimal acceleration and extrapolation parameters are ob...

A novel formulation of approximate truncated balanced realization (TBR) is introduced to unify three approaches: two iterative methods for solving the underlying Lyapunov equations – the alternating directions implicit (ADI) iteration and the rational Krylov subspace method (RKSM) – and a two-step procedure that performs a Krylov-based projection and subsequently direct TBR. The framework allow...

This paper presents the formulation, the analysis and the implementation of Alternating Direction Implicit (ADI) methods for solving the linear system of algebraic equations that arise from the discretization of multidimensional linear elliptic Partial Diierential Equations (PDEs). The theoretical analysis is carried out for a board class of PDE problems. Numerical experiments connrm the theore...

Two discrete-time orthogonal spline collocation schemes are formulated and analyzed for solving the linear time-dependent Schrr odinger equation in two space variables. These are Crank-Nicolson and alternating direction implicit (ADI) schemes employing C 1 piecewise polynomial spaces of arbitrary degree 3 in each space variable. The stability of the schemes is examined and optimal order a prior...

For large scale problems, an effective approach for solving the algebraic Lyapunov equation consists of projecting the problem onto a significantly smaller space and then solving the reduced order matrix equation. Although Krylov subspaces have been used for long time, only more recent developments have shown that rational Krylov subspaces can be a competitive alternative to the classical and v...

In this paper, an unconditionally stable three-dimensional (3-D) finite-difference time-method (FDTD) is presented where the time step used is no longer restricted by stability but by accuracy. The principle of the alternating direction implicit (ADI) technique that has been used in formulating an unconditionally stable two-dimensional FDTD is applied. Unlike the conventional ADI algorithms, ho...

A numerical method is presented to solve a two-dimensional hyperbolic diffusion problem where is assumed that both convection and diffusion are responsible for flow motion. Since direct solutions based on implicit schemes for multidimensional problems are computa-tionally inefficient, we apply an alternating direction method which is second order accurate in time and space. The stability of the...