A New Family of Optimal Eighth-Order Iterative Scheme for Solving Nonlinear Equations

The objective of this manuscript is to introduce a new family of optimal eight-order iterative methods for computing the numerical zeros of a nonlinear univariate equation that is not dependent on the second derivative. The family was designed to enhance the order of convergence by merging Bawazir’s method and Newton’s method as a third step. To demonstrate the performance of the offered scheme, assorted numerical comparisons have been investigated. In addition, the efficiency index of the new family is 1.6818

Design, Analysis, and Applications of Iterative Methods for Solving Nonlinear Systems

In this chapter, we present an overview of some multipoint iterative methods for solving nonlinear systems obtained by using different techniques such as composition of known methods, weight function procedure, and pseudo-composition, etc. The dynamical study of these iterative schemes provides us valuable information about their stability and reliability. A numerical test on a specific problem coming from chemistry is performed to compare the described methods with classical ones and to confirm the theoretical results

Solution strategies for nonlinear conservation laws

Nonlinear conservation laws form the basis for models for a wide range of physical phenomena. Finding an optimal strategy for solving these problems can be challenging, and a good strategy for one problem may fail spectacularly for others. As different problems have different challenging features, exploiting knowledge about the problem structure is a key factor in achieving an efficient solution strategy. Most strategies found in literature for solving nonlinear problems involve a linearization step, usually using Newton's method, which replaces the original nonlinear problem by an iteration process consisting of a series of linear problems. A large effort is then spent on finding a good strategy for solving these linear problems. This involves choosing suitable preconditioners and linear solvers. This approach is in many cases a good choice and a multitude of different methods have been developed. However, the linearization step to some degree involves a loss of information about the original problem. This is not necessarily critical, but in many cases the structure of the nonlinear problem can be exploited to a larger extent than what is possible when working solely on the linearized problem. This may involve knowledge about dominating physical processes and specifically on whether a process is near equilibrium. By using nonlinear preconditioning techniques developed in recent years, certain attractive features such as automatic localization of computations to parts of the problem domain with the highest degree of nonlinearities arise. In the present work, these methods are further refined to obtain a framework for nonlinear preconditioning that also takes into account equilibrium information. This framework is developed mainly in the context of porous media, but in a general manner, allowing for application to a wide range of problems. A scalability study shows that the method is scalable for challenging two-phase flow problems. It is also demonstrated for nonlinear elasticity problems. Some models arising from nonlinear conservation laws are best solved using completely different strategies than the approach outlined above. One such example can be found in the field of surface gravity waves. For special types of nonlinear waves, such as solitary waves and undular bores, the well-known Korteweg-de Vries (KdV) equation has been shown to be a suitable model. This equation has many interesting properties not typical of nonlinear equations which may be exploited in the solver, and strategies usually reserved to linear problems may be applied. In this work includes a comparative study of two discretization methods with highly different properties for this equation

Efficient upwind algorithms for solution of the Euler and Navier-stokes equations

An efficient three-dimensionasl tructured solver for the Euler and Navier-Stokese quations is developed based on a finite volume upwind algorithm using Roe fluxes. Multigrid and optimal smoothing multi-stage time stepping accelerate convergence. The accuracy of the new solver is demonstrated for inviscid flows in the range 0.675 :5M :5 25. A comparative grid convergence study for transonic turbulent flow about a wing is conducted with the present solver and a scalar dissipation central difference industrial design solver. The upwind solver demonstrates faster grid convergence than the central scheme, producing more consistent estimates of lift, drag and boundary layer parameters. In transonic viscous computations, the upwind scheme with convergence acceleration is over 20 times more efficient than without it. The ability of the upwind solver to compute viscous flows of comparable accuracy to scalar dissipation central schemes on grids of one-quarter the density make it a more accurate, cost effective alternative. In addition, an original convergencea cceleration method termed shock acceleration is proposed. The method is designed to reduce the errors caused by the shock wave singularity M -+ 1, based on a localized treatment of discontinuities. Acceleration models are formulated for an inhomogeneous PDE in one variable. Results for the Roe and Engquist-Osher schemes demonstrate an order of magnitude improvement in the rate of convergence. One of the acceleration models is extended to the quasi one-dimensiona Euler equations for duct flow. Results for this case d monstrate a marked increase in convergence with negligible loss in accuracy when the acceleration procedure is applied after the shock has settled in its final cell. Typically, the method saves up to 60% in computational expense. Significantly, the performance gain is entirely at the expense of the error modes associated with discrete shock structure. In view of the success achieved, further development of the method is proposed

A numerical method for the incompressible Navier-Stokes equations with application to two-phase flow

A finite difference numerical method is developed for the simulation of time-dependent incompressible Navier-Stokes equations As in the original projection method developed by Chorin, we first solve diffusion-convection equations to predict a intermediate velocity field which are then projected onto the space of divergence-free field. We integrate the diffusion-convection by θ implicit scheme and solve the pressure Poisson equation by successive over-relaxation method. The second order centered difference is employed to discretize both the convective and viscous terms We develop two-dimensional and three-dimensional computer programs in Fortran language Numerical results are presented We couple our inpressible Navier-Stokes solver with an interacting continuum model for two-phase flows with heat, mass transfer and phase change The Volume of Fluid method is employed in the present study Two example simulations are presented, convective melting of solid particles in a fluid under micro-gravity, and a three dimensional driven cavity problem

A four-equation two-phase flow model for sodium boiling simulation of LMFBR fuel assemblies

A three-dimensional numerical model for the simulation of sodium boiling transients has been developed. The model uses mixture mass and energy equations, while employing a separate momentum equation for each phase. Thermal equilibrium on the saturation line between coexisting phases is assumed.The four governing equations are supplemented by a number of constitutive relations, addressing the interphase and intraphase exchanges, as well as the fluid-solid interactions. It should be noted that this four-equation two-phase flow model requires only one interfacial relation, i.e., the momentum exchange, compared to the six-equation model which needs two additional relations, describing the mass and energy exchanges. Consequently, the relatively high degree of uncertainty currently associated with the interfacial exchange phenomena is considerably reduced.From a numerical point of view, the basic approach in this work is a semi-implicit method, in which pressure pulse propagation and local effects characterized by short characteristic times are treated implicitly, while convective transport and diffusion heat transfer phenomena, associated with longer time constants, are handled explicitly. The method remains tractable and efficient in multidimensional applications.Simulation of a number of experiments has yielded very encouraging results. The numerical method and the constitutive relations have performed well, especially so in light of the extreme severity of the conditions involving sodium boiling

A feed forward neural network approach for matrix computations

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.A new neural network approach for performing matrix computations is presented. The idea of this approach is to construct a feed-forward neural network (FNN) and then train it by matching a desired set of patterns. The solution of the problem is the converged weight of the FNN. Accordingly, unlike the conventional FNN research that concentrates on external properties (mappings) of the networks, this study concentrates on the internal properties (weights) of the network. The present network is linear and its weights are usually strongly constrained; hence, complicated overlapped network needs to be construct. It should be noticed, however, that the present approach depends highly on the training algorithm of the FNN. Unfortunately, the available training methods; such as, the original Back-propagation (BP) algorithm, encounter many deficiencies when applied to matrix algebra problems; e. g., slow convergence due to improper choice of learning rates (LR). Thus, this study will focus on the development of new efficient and accurate FNN training methods. One improvement suggested to alleviate the problem of LR choice is the use of a line search with steepest descent method; namely, bracketing with golden section method. This provides an optimal LR as training progresses. Another improvement proposed in this study is the use of conjugate gradient (CG) methods to speed up the training process of the neural network. The computational feasibility of these methods is assessed on two matrix problems; namely, the LU-decomposition of both band and square ill-conditioned unsymmetric matrices and the inversion of square ill-conditioned unsymmetric matrices. In this study, two performance indexes have been considered; namely, learning speed and convergence accuracy. Extensive computer simulations have been carried out using the following training methods: steepest descent with line search (SDLS) method, conventional back propagation (BP) algorithm, and conjugate gradient (CG) methods; specifically, Fletcher Reeves conjugate gradient (CGFR) method and Polak Ribiere conjugate gradient (CGPR) method. The performance comparisons between these minimization methods have demonstrated that the CG training methods give better convergence accuracy and are by far the superior with respect to learning time; they offer speed-ups of anything between 3 and 4 over SDLS depending on the severity of the error goal chosen and the size of the problem. Furthermore, when using Powell's restart criteria with the CG methods, the problem of wrong convergence directions usually encountered in pure CG learning methods is alleviated. In general, CG methods with restarts have shown the best performance among all other methods in training the FNN for LU-decomposition and matrix inversion. Consequently, it is concluded that CG methods are good candidates for training FNN of matrix computations, in particular, Polak-Ribidre conjugate gradient method with Powell's restart criteria