Implicit Iterative Scheme Research Articles

Common solutions to a finite family of inclusion problems and an infinite family of fixed point problems by a generalized viscosity implicit scheme including applications

This manuscript deals with two problems: the first one is a variational inclusion problem involving an m-accretive mapping and a finite family of inverse strongly accretive mappings, and the other one is a fixed point problem having an infinite family of strict pseudo-contraction mappings in Banach spaces. To approximate the common solution of these problems, we design a generalized viscosity implicit iterative scheme with Meir–Keeler contraction. A strong convergence result for the proposed iterative scheme is established. Applications based on convex minimization problem, linear inverse problem, variational inequality problem and equilibrium problem are derived from the main result. The numerical applicability of the main result and some applications are demonstrated by three examples. Our result extends, generalizes and unifies the previously known results given in literature.

New iterative methods for equilibrium and constrained convex minimization problems

The gradient-projection algorithm (GPA) plays an important role in solving constrained convex minimization problems. In this paper, we combine the GPA and averaged mapping approach to propose implicit and explicit composite iterative schemes for finding a common solution of an equilibrium problem and a constrained convex minimization problem. Then, we prove some strong convergence theorems which improve and extend some recent results.

Development of a fully implicit simulator for surfactant-polymer flooding by applying the variable substitution method

A perpendicular bisection (PEBI) gridding-based two-phase five-component fully implicit simulator for polymer-surfactant flooding is proposed which utilises the substitution of variable method in the numerical calculation procedure. The substitution of variable method utilises two sets of primary variables based on the phase equilibrium states. When an oil-microemulsion system exists, the primary variables to be solved are the oil phase pressure po and the volume concentrations of oil, polymer, surfactant and salt in the microemulsion, i.e., Com, Cpm, Csm and Cbm. When a single microemulsion is formed, the primary variables are po, Sm, Cpm, Csm and Cbm. A fully implicit iterative scheme is established based on the substitution of variable formulation according to oil phase disappearance or appearance which reduces the number of primary variables. The calculation speed of the new simulator is compared to a conventional fully implicit simulator, which shows that the calculation speed is significantly improved. [Received: January 2, 2017; Accepted: March 22, 2017]

Implicit iterative schemes based on singular decomposition and regularizing algorithms

Предложен новый вариант неявного метода простых итераций на основе сингулярного разложения. Показано, что данный вариант неявного метода простых итераций позволяет существенно повысить вычислительную устойчивость алгоритма и при этом обеспечивает высокую скорость его сходимости. Рассмотрено применение неявного метода простых итераций на основе сингулярного разложения для разработки итерационных регуляризирующих алгоритмов. Предлагаемые алгоритмы могут быть эффективно использованы для решения широкого класса некорректных и плохо обусловленных вычислительных задач.

Jungck-type implicit iterative algorithms with numerical examples

We introduce a new Jungck-type implicit iterative scheme and study its strong convergence, stability under weak parametric restrictions in generalized convex metric spaces and data dependency in generalized hyperbolic spaces. We show thatnewintroduced iterative scheme has better convergence rate as compared to well known Jungck implicit Mann, Jungck implicit Ishikawa and Jungck implicit Noor iterative schemes. It is also shown that Jungck implicit iterative schemes converge faster than the corresponding Jungck explicit iterative schemes. Validity of our analytic proofs is shown through numerical examples. Our results are improvements and generalizations of some recent results of Khan et al.[21], Chugh et al.[8] and many others in fixed point theory.

Approximating common fixed point of asymptotically nonexpansive mappings without convergence condition

In the context of a hyperbolic space, we introduce and study convergence of an implicit iterative scheme of a finite family of asymptotically nonexpansive mappings without convergence condition. The results presented substantially improve and extend several well-known resullts in uniformly convex Banach spaces.

About one splitting scheme for the nonlinear problem of thermal convection

The paper is devoted to the questions of mathematical modeling of heating processes in hydrodynamics, namely to development and to mathematical justification of numerical algorithms for solving three-dimensional equations of free convection in natural variables. The purpose is to research implicit iterative schemes for the numerical solution of Boussinesq-type fixed (stationary) equations.The research uses mathematical modeling, mathematical programming, the Visual Fortran programming language, and the Axum 7.0 graphics program. Computational mathematics and functional analysis methods are used for the mathematical argumentation of iterative algorithms.The questions of convergence and estimate of a degree of convergence of one nonlinear splitting algorithm are considered, made for the difference analogues of the system of free convection steady-state equations in variables “velocity vector and pressure”, written to shifted grids with symmetric approximation.The implicit iterative splitting algorithms for the difference analogues of the system of free convection steady-state equations in variables “velocity vector and pressure” are considered, written to shifted grids with symmetric approximation. The problems of stability of the difference problems according to the initial data and the right member, convergence and estimate of the linear algorithm degree of convergence were studied.The results of this research can be useful in studies on difference schemes for hydrodynamic equations; they can also be used to further develop the theory of numerical solution of mathematical physics problems.The research results may be used in information system development for the automation of heat-aggregation exchange problem solving and as a teaching material for students learning mathematics, mechanics, and IT technologies.

Numerical solutions of nanofluid bioconvection due to gyrotactic microorganisms along a vertical wavy cone

An analysis is performed to study the bioconvection flow with heat and mass transfer of nanofluid containing gyrotactic microorganisms along a vertical wavy cone. The model includes equations expressing conservation of mass, momentum, thermal energy, nanoparticles and microorganisms. Primitive variable formulations (PVF) are used to transform the dimensionless boundary layer equations into a convenient coordinate system. The equations obtained from PVF are integrated numerically via an implicit finite difference iterative scheme. The effect of the controlling parameters on the dimensionless quantities such as local Nusselt number, local Sherwood number and local density number of motile microorganisms are explored. The numerical results show that the amplitude of the wavy surface of the cone and half cone angle has pronounced effect on the heat transfer coefficient, mass transfer coefficient and density number of the microorganisms coefficient.

An implicit Lie-group iterative scheme for solving the nonlinear Klein–Gordon and sine-Gordon equations

In this article, the nonlinear Klein–Gordon and sine-Gordon equations are solved by pondering the semi-discretization numerical schemes and then, the resulting ordinary differential equations at the discretized spaces are numerically integrated toward the time direction by using the implicit Lie-group iterative method to find the unknown physical quantity. When six numerical experiments are examined, we reveal that the present implicit Lie-group iterative scheme is applicable to the nonlinear Klein–Gordon and sine-Gordon equations and convergent very fast at each time marching step, and the accuracy is raised several orders, of which the numerical results are rather accurate, effective and stable.

Numerical Simulation of Heavy Crude Oil Combustion in Porous Combustion Tube

This work presents a compositional numerical model to simulate the combustion of heavy crude oil in a porous combustion tube. The combustion tube experiments replicate an enhanced oil recovery process by in-situ combustion (ISC) of crude oil in a petroleum reservoir. Due to the economic constrains associated with the combustion tube tests, numerical simulation has been identified as an efficient and economical tool in understanding the complex ISC process. In the present model, mass and energy conservations have been drawn considering an advective, diffusive, and reactive flow of multiple components in multi-phase. The model includes temperature- and pressure-dependent fluid properties and considers the effect of capillary pressure. A block-centered finite-difference-based numerical tool with an implicit quasi-Newton iterative solver scheme has been developed to solve the conservation equations. The numerical model is validated with the combustion-tube results reported by Fadaei et al. (2011). The combustion front profile predicted by the present numerical model has been found to be in good agreement with the reported results. The numerical results projected a cumulative oil recovery of about 60% of original oil in place. The combustion front has been found to propagate with an average peak temperature of 715 K.

APPLICATIONS, THEORETICAL ANALYSIS, NUMERICAL METHOD, AND MECHANICAL MODEL OF THE POLYMER FLOODING IN POROUS MEDIA

Applications, theoretical analysis, and numerical methods are introduced for the simulation of mechanical models of the flow in porous media of polymer flooding in this paper. Considering petroleum geology, geochemistry, computational permeation fluid mechanics, and computer technology, we state the models of seepage mechanics and put forward a sequence of implicit upwind difference iteration schemes based on refined fractional steps of the upstream. This scheme can compute the pressure, the saturation, and the concentrations of different chemical components. A type of software applied in major industries has been completed and carried out in numerical analysis and simulation of oil extraction in the Daqing Oil Field and Shengli Oil Field, which gives rise to huge economic benefit and social benefit. This software has the high-order accuracy in simulation and other characteristics as follows. A spatial step can be taken by 10 meters, and the number of nodes is up to about 100,000. Precise analysis is stated for a simplified mathematical model of this type and it provides a powerful tool to solve the international famous problem.

Viscosity approximation methods for two nonexpansive semigroups in CAT(0) spaces

The purpose of this paper is by using the viscosity approximation method to study the strong convergence problem for two one-parameter continuous semigroups of nonexpansive mappings in CAT(0) spaces. Under suitable conditions, some strong convergence theorems for the proposed implicit and explicit iterative schemes to converge to a common fixed point of two one-parameter continuous semigroups of nonexpansive mappings are proved, which is also a unique solution of some kind of variational inequalities. The results presented in this paper extend and improve the corresponding results of some others.

Convergence and stability of balanced methods for stochastic delay integro-differential equations

This paper deals with a family of balanced implicit methods for the stochastic delay integro-differential equations. It is shown that the balanced methods, which own the implicit iterative scheme in the diffusion term, give strong convergence rate of at least 1/2. It proves that the mean-square stability for the stochastic delay integro-differential equations is inherited by the strong balanced methods and the weak balanced methods with sufficiently small stepsizes. Several numerical experiments are given for illustration and show that the fully implicit methods are superior to those of the explicit methods in terms of mean-square stabilities.

Coupled Navier–Stokes/Direct Simulation Monte Carlo Simulation of Multicomponent Mixture Plume Flows

A coupled particle-continuum simulation method is presented to compute the multicomponent mixture plume flow from the attitude-control engines of a satellite through the use of the Navier–Stokes solver and direct simulation Monte Carlo method. In this study, the implicit hybrid flux iteration scheme for the axisymmetric compressible Navier–Stokes equations is constructed to solve the inner flowfield of the nozzle. The Navier–Stokes computing technique is implemented considering slip boundary theory for the near-continuum slip flow near the nozzle exit. Direct simulation Monte Carlo methods for the axisymmetric core plume and three-dimensional far-field plume flows of multispecies mixture gas are developed by using a radial weighted factor, by tracking the molecular motion trajectory, and with secondary Cartesian and unstructured cell processing techniques. A multiregion decomposition and hybrid Navier–Stokes/direct simulation Monte Carlo algorithm with two-way and one-way coupling is established to solve the internal and external flow of the thruster, including the near-field, far-field, backflow, and gas-surface contaminated regions. After constructing the coupled Navier–Stokes/DSMC simulation scheme, the present method is applied to solve the plume flowfield and impinging contamination effects from the satellite attitude-control engine and solar array panel; the simulation results correspond well with the experimental measurements from the low-density wind tunnel and the theoretical predictions. To study the contamination effects produced by the multicomponent mixture plume, the current method is employed to simulate a five-component mixture plume flowfield of tens of meters from two representative attitude-control engines installed in different locations of a satellite in orbit. The results of the nonreacting multicomponent flow with equivalent mole fractions can be achieved with a one-component gas simulation if the species is assigned the properties of the mixture. The deposition rate of the plume flows produced by the two teamwork engines can be superimposed on each other. This method can be used to efficiently predict the impingement contamination effects of the gas-fired mixture plume flow on the satellite and solar array panels.

A transformation method for solving Hamilton-Jacobi-Bellman equation for constrained dynamic stochastic optimal allocation problem

In this paper we propose and analyze a method based on the Riccati transformation for solving the evolutionary Hamilton-Jacobi-Bellman equation arising from the stochastic dynamic optimal allocation problem. We show how the fully nonlinear Hamilton-Jacobi-Bellman equation can be transformed into a quasi-linear parabolic equation whose diffusion function is obtained as the value function of certain parametric convex optimization problem. Although the diffusion function need not be sufficiently smooth, we are able to prove existence, uniqueness and derive useful bounds of classical H\"older smooth solutions. We furthermore construct a fully implicit iterative numerical scheme based on finite volume approximation of the governing equation. A numerical solution is compared to a semi-explicit traveling wave solution by means of the convergence ratio of the method. We compute optimal strategies for a portfolio investment problem motivated by the German DAX 30 Index as an example of application of the method. doi:10.1017/S144618111300031X

Convergence theorems of a new iteration for two nonexpansive mappings

The purpose of this paper is to introduce the following new general implicit iteration scheme for approximating the common fixed points of a pair of nonexpansive mappings in a uniformly convex Banach space: for any , the iterative process defined by , , where , , , , , , are seven sequences of real numbers satisfying , , and are two nonexpansive mappings. We approximate the common fixed points of these two mappings by weak and strong convergence of the scheme.

Viscosity Approximation Methods for a Family of Nonexpansive Mappings in CAT(0) Spaces

The purpose of this paper is using the viscosity approximation method to study the strong convergence problem for a family of nonexpansive mappings in CAT(0) spaces. Under suitable conditions, some strong convergence theorems for the proposed implicit and explicit iterative schemes to converge to a common fixed point of the family of nonexpansive mappings are proved which is also a unique solution of some kind of variational inequalities. The results presented in this paper extend and improve the corresponding results of some others.

Theory and Application of Numerical Simulation of Chemical Flooding in High Temperature and High Salt Reservoirs

Applications, theoretical analysis and numerical methods are introduced for the simulation of mechanical models and principles of the porous flow in high temperature, high salt, complicated geology and large-scale reservoirs in this paper. Considering petroleum geology, geochemistry, computational permeation fluid mechanics and computer technology, we state the models of permeation fluid mechanics and put forward a sequence of implicit upwind difference iteration schemes based on refined fractional steps of the upstream, which can compute the pressures, the saturation and the concentrations of different chemistry components. A type of software applicable in major industries has been completed and carried out in numerical analysis and simulations of oil extraction in Shengli Oil-field, which brings huge economic benefits and social benefits. This software gives many characters: spatial steps are taken as ten meters, the number of nodes is up to hundreds of thousands and simulation time period can be tens of years and the high-order accuracy can be promised in numerical data. Precise analysis is present for simplified models of this type and that provides a tool to solve the international famous problem.

Iterative Fixed-Point Methods for Solving Nonlinear Problems: Dynamics and Applications

and Applied Analysis Volume 2014, Article ID 313061, 2 pages http://dx.doi.org/10.1155/2014/313061 2 Abstract and Applied Analysis α and c whose associated iterative schemes show stable and unstable behavior. W. Zhou and J. Kou present a third-order variant of Potra-Ptak’s iterative method for solving nonlinear systems in Banach spaces, by using a new vector extrapolation technique, whose local and semilocal convergence are proved. In their manuscript, M. J. Martinez et al. use the fixed point iterative scheme to correct the errors between Hipparcos and ICRF2 Celestial catalogues.This allows the authors to link both systems, giving a uniform reference regardless the celestial body (an astronomical radio source) is visible or not. In the manuscript “Existence and Algorithm for the System of Hierarchical Variational Inclusion Problems,” the authors study the existence and approximation of solution of this kind of problems in Hilbert spaces. By using Mainge’s approach, N. Wairojjana and P. Kumam improve and generalize some known results in this area. In the context of iterative methods for solving nonlinear matrix equations, four papers have been presented. M. K. Razavi et al. show a new iterative method for computing the approximate inverse of a nonsingular matrix. Moreover, F. Soleymani et al. describe an iterative process with cubical rate of convergence for finding the principal matrix square root. Application of this approach is illustrated by solving a matrix differential equation. On the other hand, in “An Algorithm for Computing Geometric Mean of Two Hermitian Positive Definite Matrices via Matrix Sign,” the authors present a fast algorithm for computing the geometric mean of twomatrices of this specific type. Finally, in “Approximating the Matrix Sign Function Using a Novel Iterative Method,” the authors present an iterative method for finding the sign of a square complex matrix. L. Guo et al. use the reproducing kernel iterative method to solve a class of two-point boundary value problems. By homogenizing the boundary conditions, the two-point boundary value problems are converted into the equivalent nonlinear operator equation. They prove that both solutions are equivalent and show the analysis of error and the numerical stability of the method. In “Existence of Solutions for a Coupled Systemof Second and Fourth Order Elliptic Equations,” the author generalizes quasilinearization technique to obtain a monotone sequence of iterates converging uniformly and quadratically to a solution of this kind of systems. Finally, in “TheHybrid Steepest DescentMethod for Split Variational Inclusion and Constrained ConvexMinimization Problems,” the authors introduce new implicit and new explicit iterative schemes for finding a common element of the set of solutions of a constrain convex minimization problems and the set of solutions of the split variational inclusion problem.

Composite Implicit Iteration Process for Two Finite Families of Generalized Asymptotically Quasi-Nonexpansive Mappings

In this article, A demiclosed principle is proved for generalized asymptotically nonexpansive mapping in a Banach space. Additionally, it proves strong convergence of a composite implicit iteration scheme to a common fixed point for two finite families of generalized asymptotically quasi-nonexpansive mappings in a nonempty closed convex subset of a uniformly convex Banach space. We derive a necessary and sufficient condition for the strong convergence of this iteration process to a common fixed point of these mappings. The results of this article improve and extend the corresponding results of Xu and Ori [23], Zhou and Chang [26], Chang et al. [3], Yang and Yu [25], Shahzad and Zegeye [20].