Preconditioned conjugate gradient methods for the solution of indefinite least squares problems

The conjugate gradient (CG) method is considered for solving the large and sparse indefinite least squares (ILS) problem min  _x (b−Ax) ^T J(b−Ax) where J=diag (I _p ,−I _q ) is a signature matrix. However the rate of convergence becomes slow for ill-conditioned problems. The QR-based preconditioner is found to be effective in accelerating the convergence. Numerical results show that the sparse Householder QR-based preconditioner is superior to the CG method especially for sparse and ill-conditioned problems.     

Preconditioned iterative methods for the numerical solution of Fredholm equations of the second kind

In this paper, we consider the numerical solution of Fredholm integral equations of the second kind:

Application of spectral methods to the solution of Navier-Stokes equations

In this paper a number of topics that arise in application of Chebyshev expansion methods to the solution of the Navier-Stokes equations are addressed. These include the equivalence of finite differences and finite elements approaches, a new velocity-pressure formulation that permits easy extension to three dimensions, evaluation of the importance of pressure boundary conditions and the virtues of collocation over the tau method for satisfying boundary conditions. Example results from the velocity-pressure formulation which eliminate a non-linear momentum equation in favor of the linear continuity equation are presented. The results are for a 2-D unsteady flow on a flat plate at large Reynolds numbers. The behavior of an unsteady disturbance in such a flow is examined and compared with previous stream-function vorticity results of Murdock.     

Iterative methods with spectral preconditioning for elliptic equations

We derive two preconditioners for the iterative solution of the linear system arising from Chebyshev approximation of a generalized Helmholtz problem. These preconditioners are constructed as full spectral approximations of a differential problem close in some sense to the original one. The analysis and numerical experiments show the efficiency of these iterative schemes and indicate that they appear as valuable alternative to the usual finite difference or finite element preconditionings.     

Preconditioned rotated iterative methods in the solution of elliptic partial differential equation

A second-order finite difference scheme derived from rotated discretisation formula is employed in conjunction with a preconditioner to obtain highly accurate and fast numerical solution of the two-dimensional elliptic partial differential equation. The use of a ‘splitting’ preconditioning strategy will be shown to improve the spectral properties of the matrix of the linear system resulting from this discretisation by minimising the eigenvalue spectrum of the transformed matrix. The application of this technique to several acceleration iterative methods, such as Simultaneous displacement, Richardson's and Chebyshev accelerated methods, are presented and discussed.     

Non-polynomial cubic spline methods for the solution of parabolic equations

Second-order parabolic partial differential equations are solved by using a new three level method based on non-polynomial cubic spline in the space direction and finite difference in the time direction. Stability analysis of the method has been carried out and we have shown that our method is unconditionally stable. It has been shown that by suitably choosing the parameters most of the previous known methods for homogeneous and non-homogeneous cases can be obtained from our method. We also obtain a new high accuracy scheme of O(k ⁴+h ⁴). Numerical examples are given to illustrate the applicability and efficiency of the new method.     

Iterative methods for the solution of the navier equations of elasticity

The numerical solution of the Navier equations discretized by finite elements is studied by various forms of pre-conditioned conjugate gradient methods. In particular, the dependence of the number of iterations is examined as a function of Poisson's ratio. A comparison is made with direct solution methods, and the dependence of the discretization error on Poisson's ratio is also discussed.     

Fast solution of implicit methods for linear hyperbolic equations

A factorisation method is described for the fast numerical solution of constant tridiagonal Toeplitz linear systems which occur repeatedly in the solution of the implicit finite difference equations derived from linear first order hyperbolic equations, i.e. the Transport equation, under a variety of boundary conditions. In this paper, we show that such special linear systems can be solved efficiently by the factorisation of the coefficient matrix into two easily inverted matrices.     

Extended one-step methods for the numerical solution of ordinary differential equations

A class of one-step finite difference formulae for the numerical solution of first-order differential equations is considered. The accuracy and stability properties of these methods are investigated. By judicious choice of the coefficients in these formulae a method is derived which is both A-stable and third-order convergent. Moreover the new method is shown to be L-stable and so is appropriate for the solution of certain stiff equations. Numerical results are presented for several test problems.     

Strong and weak Lagrange-Galerkin spectral element methods for the shallow water equations

The Lagrange-Galerkin spectral element method for the two-dimensional shallow water equations is presented. The equations are written in conservation form and the domains are discretized using quadrilateral elements.Lagrangian methods integrate the governing equations along the characteristic curves, thus being well suited for resolving the nonlinearities introduced by the advection operator of the fluid dynamics equations.Two types of Lagrange-Galerkin methods are presented: the strong and weak formulations. The strong form relies mainly on interpolation to achieve high accuracy while the weak form relies primarily on integration. Lagrange-Galerkin schemes offer an increased efficiency by virtue of their less stringent CFL condition. The use of quadrilateral elements permits the construction of spectral-type finite-element methods that exhibit exponential convergence as in the conventional spectral method, yet they are constructed locally as in the finite-element method; this is the spectral method.In this paper, we show how to fuse the Lagrange-Galerkin methods with the spectral element method and present results for two standard test cases in order to compare and contrast these two hybrid schemes.     

Two modified spectral conjugate gradient methods and their global convergence for unconstrained optimization

In this paper, two modified spectral conjugate gradient methods which satisfy sufficient descent property are developed for unconstrained optimization problems. For uniformly convex problems, the first modified spectral type of conjugate gradient algorithm is proposed under the Wolfe line search rule. Moreover, the search direction of the modified spectral conjugate gradient method is sufficiently descent for uniformly convex functions. Furthermore, according to the Dai–Liao's conjugate condition, the second spectral type of conjugate gradient algorithm can generate some sufficient decent direction at each iteration for general functions. Therefore, the second method could be considered as a modification version of the Dai–Liao's algorithm. Under the suitable conditions, the proposed algorithms are globally convergent for uniformly convex functions and general functions. The numerical results show that the approaches presented in this paper are feasible and efficient.     

Preconditioned iterative methods for the large sparse symmetric eigenvalue problem

In general, two strategies are used in methods for the solution of large sparse eigensystems. The former are transformation and elimination methods which may change the structure of the original matrix, destroy sparsity and are only suitable if all or most of the eigenvalues are required. The latter includes methods which are iterative and no change in the structure of the original matrix occurs. These methods are often employed when one or several of the eigenvalues are required. In this paper we study iterative methods whereby the extreme eigenvalues and their corresponding eigenvectors are evaluated by a new preconditioning method which with a suitable choice of shift of origin and preconditioning parameter produces a powerful convergent method to cope with problems in this class.     

Analysis and application of a parallel spectral element method for the solution of the Navier-Stokes equations

We present an analysis of the parallel spectral element method for solution of the unsteady incompressible Navier-Stokes equations in general three-dimensional geometries. The approach combines high-order spatial discretizations with iterative solution techniques in a way which exploits with high efficiency the currently available medium-grained distributed-memory parallel computers. Measured performance analysis on the Intel vector hypercubes and example Navier-Stokes calculations demonstrate that parallel processing can now be considered an effective fluid mechanics analysis tool.     

A comparison of numerical methods for the solution of quasilinear partial differential equations

Various methods of approximating a parabolic partial differential equation by a system of ordinary differential equations are discussed. The methods are compared using the results of numerical experiments with a highly efficient integration procedure designed for this type of problem.     

Preconditioned methods for simulations of low speed compressible flows

A time-derivative preconditioned system of equations suitable for the numerical simulation of inviscid compressible flow at low speeds is formulated. The preconditioned system of equations are hyperbolic in time and remain well-conditioned in the incompressible limit. The preconditioning formulation is easily generalized to multicomponent/multiphase mixtures. When applying conservative methods to multicomponent flows with sharp fluid interfaces, nonphysical solution behavior is observed. This stimulated the authors to develop an alternative solution method based on the nonconservative form of the equations which does not generate the aforementioned nonphysical behavior. Before the results of the application of the nonconservative method to multicomponent flow problems is reported, the accuracy of the method on single component flows will be demonstrated. In this report a series of steady and unsteady inviscid flow problems are simulated using the nonconservative method and a well-known conservative scheme. It is demonstrated that the nonconservative method is both accurate and robust for smooth low speed flows, in comparison to its conservative counterpart.     

Adams methods for the efficient solution of stochastic differential equations with additive noise

The application of Adams methods for the numerical solution of stochastic differential equations is considered. Especially we discuss the path-wise (strong) solutions of stochastic differential equations with additive noise and their numerical computation. The special structure of these problems suggests the application of Adams methods, which are used for deterministic differential equations very successfully. Applications to circuit simulation are presented.