Development of iterative techniques for the solution of unsteady compressible viscous flows

The development of efficient iterative solution methods for the numerical solution of two- and three-dimensional compressible Navier-Stokes equations is discussed. Iterative time marching methods have several advantages over classical multi-step explicit time marching schemes, and non-iterative implicit time marching schemes. Iterative schemes have better stability characteristics than non-iterative explicit and implicit schemes. In this work, another approach based on the classical conjugate gradient method, known as the Generalized Minimum Residual (GMRES) algorithm is investigated. The GMRES algorithm has been used in the past by a number of researchers for solving steady viscous and inviscid flow problems. Here, we investigate the suitability of this algorithm for solving the system of non-linear equations that arise in unsteady Navier-Stokes solvers at each time step

Non-iterative and exact method for constraining particles in a linear geometry

We present a practical numerical method for evaluating the Lagrange multipliers necessary for maintaining a constrained linear geometry of particles in dynamical simulations. The method involves no iterations, and is limited in accuracy only by the numerical methods for solving small systems of linear equations. As a result of the non-iterative and exact (within numerical accuracy) nature of the procedure there is no drift in the constrained geometry, and the method is therefore readily applied to molecular dynamics simulations of, e.g., rigid linear molecules or materials of non-spherical grains. We illustrate the approach through implementation in the commonly used second-order velocity explicit Verlet method.Comment: 12 pages, 2 figure

A Multi-Domain Spectral Collocation Approach for Solving Lane-Emden Type Equations

In this work, we explore the application of a novel multi-domain spectral collocation method for solving general non-linear singular initial value differential equations of the Lane-Emden type. The proposed solution approach is a simple iterative approach that does not employ linearisation of the differential equations. Spectral collocation is used to discretise the iterative scheme to form matrix equations that are solved over a sequence of non-overlapping sub-intervals of the domain. Continuity conditions are used to advance the solution across the non-overlapping sub-intervals. Different Lane-Emden equations that have been reported in the literature have been used for numerical experimentation. The results indicate that the method is very effective in solving Lane-Emden type equations. Computational error analysis is presented to demonstrate the fast convergence and high accuracy of the method of solution

Linear iterative solvers for implicit ODE methods

The numerical solution of stiff initial value problems, which lead to the problem of solving large systems of mildly nonlinear equations are considered. For many problems derived from engineering and science, a solution is possible only with methods derived from iterative linear equation solvers. A common approach to solving the nonlinear equations is to employ an approximate solution obtained from an explicit method. The error is examined to determine how it is distributed among the stiff and non-stiff components, which bears on the choice of an iterative method. The conclusion is that error is (roughly) uniformly distributed, a fact that suggests the Chebyshev method (and the accompanying Manteuffel adaptive parameter algorithm). This method is described, also commenting on Richardson's method and its advantages for large problems. Richardson's method and the Chebyshev method with the Mantueffel algorithm are applied to the solution of the nonlinear equations by Newton's method

The Analysis of Iterative Elliptic PDE Solvers Based on The Cubic Hermite Collocation Discretization

Abstract. Collocation methods based on bicubic Hermite piecewise polynomials have been proven effective t.echniques for solving second-order linear elliptic PDEs with mixed boundary conditions. The corresponding linear system is in general non-symmetric and non-diagonally dominant. Iterative methods for their solution arc not known and they aTC currently solved using Gauss elimination with scaling and partial pivoting. Point iterat.ive methods do not convcrge even for the collocation equations obtained from model PDE problems. The del/elopment of efficient iterative solvers for these equations is necessary for three-dimensional problems and their parallel solution, since direct solvers tend to be space bound and their parallelization is difficult. In this thesis, we develop block iterative methods for the collocation equations of elliptic PDEs defined on a rectangle and subject to uncoupled mixed boundary conditions. For model problems of this type, we derive analytic expressions for the eigenvalues of the block Jacobi iteration matrix: and determine the optimal parameter for the block SOR method. For the case of general domains, the iterative solution of tile collocation equations is still an open problem. We address this open problem b

Learning Output Kernels for Multi-Task Problems

Simultaneously solving multiple related learning tasks is beneficial under a variety of circumstances, but the prior knowledge necessary to correctly model task relationships is rarely available in practice. In this paper, we develop a novel kernel-based multi-task learning technique that automatically reveals structural inter-task relationships. Building over the framework of output kernel learning (OKL), we introduce a method that jointly learns multiple functions and a low-rank multi-task kernel by solving a non-convex regularization problem. Optimization is carried out via a block coordinate descent strategy, where each subproblem is solved using suitable conjugate gradient (CG) type iterative methods for linear operator equations. The effectiveness of the proposed approach is demonstrated on pharmacological and collaborative filtering data