Successive over-relaxation

The primary goal of this work was to study advantages of numerical methods used for the creation of continuous time Markov chain models (CTMC) of voltage gating of gap junction (GJ) channels composed of connexin protein. This task was accomplished by describing gating of GJs using the formalism of the stochastic automata networks (SANs), which allowed for very efficient building and storing of infinitesimal generator of the CTMC that allowed to produce matrices of the models containing a distinct block structure. All of that allowed us to develop efficient numerical methods for a steady-state solution of CTMC models. This allowed us to accelerate CPU time, which is necessary to solve CTMC models, ∼20 times.

The landfill construction has caused many negative impacts on the surrounding environment, particularly groundwater. Evaluation of the function of the leachate collection pipe at the landfill site is indispensable for managing the landfill operation. 3D groundwater flow simulation may be applicable but it requires much capacity of computer and time consumption comparing with 2D groundwater flow simulation due to the huge calculations. Therefore, the 2D horizontal groundwater flow simulation (2D h ) was carried out. However, the most difficulty is the assignment of the groundwater head at the leachate collection pipe buried for leachate drainage. Therefore, in the present paper the 2D vertical model (2D v ) was applied to calculate the change of groundwater table above the leachate collection pipe. This paper paid attention to examine the validation of the assignment of the leachate collection pipe boundary by applying the results of the 2D v . The 2D h was coupled with the rainwater recharge model to solve the partial differential equation of groundwater flow. Finite difference method and iterative successive over relaxation were applied. The drainage volume of leachate collection was summed up in the whole landfill site and compared with the average volume of treated waste water. The study demonstrated that the groundwater level at the vicinity of the drainage pipe in the 2D v analysis is reasonably assigned for the 2D h .

In this paper, refinement of generalized accelerated over relaxation (RGAOR) iterative method is presented based on the Nekrassov-Mehmke 1- method (NM1) procedure for solving system of linear equations of the form , where is a nonsingular real matrix of order , is a given dimensional real vector. The coefficient matrix is split as in , where is a banded matrix of band width and and are strictly lower and strictly upper triangular parts of the matrix respectively. The finding shows that the iterative matrix of the new method is the square of generalized accelerated successive over relaxation iterative matrix. The convergence of the new method is studied and few numerical examples are considered to show the efficiency of the proposed methods. As compared to generalized accelerated successive over relaxation (SOR2GNM1, SOR1GNM1), the results reveal that the present method (RSOR1GNM1, RSOR2GNM1) converges faster and its error at any predefined error of tolerance is less than the other m...

The fundamental problem of linear algebra is to solve the system of linear equations (SOLE's). To solve SOLE's, is one of the most crucial topics in iterative methods. The SOLE's occurs throughout the natural sciences, social sciences, engineering, medicine and business. For the most part, iterative methods are used for solving sparse SOLE's. In this research, an improved iterative scheme namely, ''a new improved classical iterative algorithm (NICA)'' has been developed. The proposed iterative method is valid when the co-efficient matrix of SOLE's is strictly diagonally dominant (SDD), irreducibly diagonally dominant (IDD), M-matrix, Symmetric positive definite with some conditions and H-matrix. Such types of SOLE's does arise usually from ordinary differential equations (ODE's) and partial differential equations (PDE's). The proposed method reduces the number of iterations, decreases spectral radius and increases the rate of convergence. Some numerical examples are utilized to demonstrate the effectiveness of NICA over Jacobi (J), Gauss Siedel (GS), Successive Over Relaxation (SOR), Refinement of Jacobi (RJ), Second Refinement of Jacobi (SRJ), Generalized Jacobi (GJ) and Refinement of Generalized Jacobi (RGJ) methods.

The main objective for this study is to examine the efficiency of block iterative method namely Four-Point Explicit Group Successive Over Relaxation (4EGSOR) iterative method. The nonlinear Burger's equation is then solved through the application of nonlocal arithmetic mean discretization (AMD) scheme to form a linear system. Next, to scrutinize the efficiency of 4EGSOR with Gauss-Seidel (GS) and Successive Over Relaxation (SOR) iterative methods, the numerical experiments for four proposed problems are being considered. By referring to the numerical results obtained, we concluded that 4EGSOR is more superior than GS and SOR iterative methods in aspects of number of iterations and execution time.

In this paper, the Burger's equations have been approximated by using the second-order finite difference scheme and the half-sweep nonlocal arithmetic discretization scheme to construct the half-sweep generated linear system. Then, we investigate the applicable formulation of Half-sweep SuccessiveOver Relaxation (HSSOR) iterative method for solving this linear system. In order to verify the effectiveness of the HSSOR iterative method, this paper also included the Fullsweep Successive OverRelaxation (FSSOR) and Full-Sweep Gauss-Seidel (FSGS) iterative methods. The performance analysis of these three proposed iterative methods is illustrated by solving four proposed Burger's problems. The numerical results illustrate the great performance of the HSSOR iterative method together with half-sweep nonlocal arithmetic discretization scheme to solve the Burger's equations in senses of execution time and number of iterations.

The problem of options gets a special importance due to the fact that many of the managerial decisions can be assimilated to options. The paper deals with numerical methods of financial options evaluation. The mathematical model of determining the values of an option of European type is partial differential equation with an initial condition and two boundaries conditions, and for its solving, finite differences methods can be used. Out of these methods, the Crank-Nicolson method is proposed to be used. As Crank-Nicolson method is an implicit one, applying it leads to a linear system of equations, for whose solving the LU algorithm, Gauss method, Jacobi method, Gauss -Seidel method, method of successive over-relaxations are used. Keyworks: European options, values of an option, Crank -Nicolson Method, LU algorithm, Gauss method, Jacobi method, Gauss -Seidel method, method of successive over-relaxations. Introducere 1.1. Opţiuni financiare O opţiune conferă deţinătorului dreptul, dar înlătură obligaţia, de a tranzacţiona o cantitate oarecare dintr-un activ financiar, la o dată fixată înainte şi la un preţ prestabilit. Cumpărătorul unei opţiuni are dreptul, însă nu şi obligaţia, de a cumpăra sau vinde o anumită cantitate de active suport (materiale, financiare, valutare etc.), la o dată convenită şi la un preţ fixat înainte. Vânzătorul unei opţiuni îşi asumă irevocabil obligaţia de a vinde sau cumpăra o cantitate de active, la termenul de scadenţă, la un preţ convenit, chiar dacă la momentul exercitării opţiunii piaţa îi este nefavorabilă. După data expirării întâlnim opţiuni de tip european -care se exercită doar la exerciţiu şi opţiuni de tip american -care se pot exercita în orice moment de timp anterior scadenţei. Din punct de vedere al naturii există opţiuni de tip call (de cumpărare) şi opţiuni de tip put (de vânzare). Pentru evaluarea unei opţiuni europene este necesar să se rezolve o ecuaţie cu derivate parţiale cu o condiţie iniţială şi două condiţii la limită.

The aim of this paper is to examine the effectiveness of Half-Sweep Successive Over Relaxation (HSSOR) method with nonlocal discretization scheme which is derived based on the four-point rotated nonlocal arithmetic mean scheme in solving nonlinear elliptic boundary value problems. By using an approximate equation based on the second order finite difference scheme, the half-sweep approximation equation has been derived. Then, the nonlocal discretization scheme is applied to transform the system of nonlinear approximation equations into the corresponding system of linear equations. Throughout numerical results, it can be pointed out that the proposed HSSOR method was superior in terms of number of iterations, execution time and maximum error compared to Full-Sweep Successive Over-relaxation (FSSOR) and Half-Sweep Gauss-Seidel (HSGS).

In order to solve the large sparse systems of linear equations arising from numerical solutions of twodimensional steady incompressible viscous flow problems in primitive variable formulation, Ran and Yuan [On modified block SSOR iteration methods for linear systems from steady incompressible viscous flow problems, Appl. Math. Comput. 217 (2010), pp. 3050-3068] presented the block symmetric successive over-relaxation (BSSOR) and the modified BSSOR iteration methods based on the special structures of the coefficient matrices. In this study, we present the modified alternating direction-implicit (MADI) iteration method for solving the linear systems. Under suitable conditions, we establish convergence theorems for the MADI iteration method. In addition, the optimal parameter involved in the MADI iteration method is estimated in detail. Numerical experiments show that the MADI iteration method is a feasible and effective iterative solver.

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Highlights • We propose a variant of the relaxation step for iterative solvers. • This variant improves the convergence for matrices with real eigenvalues. • The proposed scheme profitably applies to elliptic problems.

In order to understand the effects of the landfill operation on groundwater flow behavior, 2D horizontal groundwater simulation model was carried out. The model saved the memory of computer and time consumption comparing with 3D groundwater flow model. However, the most difficulty is the assignment of collecting pipe boundary in the study site. Therefore, 2D vertical model was applied to calculate the change of groundwater table above the collecting pipe. This paper paid attention to examine the validation of the assignment of the collecting pipe boundary by applying the results of 2D vertical model. 2D horizontal model was coupled with the recharge model to solve the partial differential equation of groundwater flow. Finite difference method and iterative successive over relaxation were applied. The drainage volume of leachate collection was summed up in the whole landfill site and compared with the average volume of treated waste water. The study demonstrated that the results of 2D vertical model validated and can be applied to 2D horizontal groundwater flow simulation.

This paper considers a batch gradient method with ⁄ regularization for Pi-sigma neural networks. In origin, by introducing an ⁄ regularization term involves absolute value and is not differentiable into the error function. A key point of this paper, specifically, the smoothing ⁄ regularization is a term proportional to the norm of the weights. The role of the smoothing ⁄ regularization term is to control the magnitude of the weights and to improve the generalization performance of the networks. The weights are proved to be bounded during the training process, thus the conditions that are required for convergence analysis of batch gradient method in literature are simplified.

Given any linear stationary iterative methods in the form z^(i+1)=Jz^(i)+f, where J is the iteration matrix, a significant improvements of the iteration matrix will decrease the spectral radius and enhances the rate of convergence of the particular method while solving system of linear equations in the form Az=b. This motivates us to refine the Extended Accelerated Over-Relaxation (EAOR) method called Refinement of Extended Accelerated Over-Relaxation (REAOR) so as to accelerate the convergence rate of the method. In this paper, a refinement of Extended Accelerated Over-Relaxation method that would minimize the spectral radius, when compared to EAOR method, is proposed. The method is a 3-parameter generalization of the refinement of Accelerated Over-Relaxation (RAOR) method, refinement of Successive Over-Relaxation (RSOR) method, refinement of Gauss-Seidel (RGS) method and refinement of Jacobi (RJ) method. We investigated the convergence of the method for weak irreducible diagonally...

The transient buoyancy-driven convection in a water saturated porous cavity with internal heat generation is studied numerically. The Brinkman-Forchheimer-extended Darcy model is employed to investigate the average heat transfer rate and to study the effects of density maximum, the Grashof number, porosity, the Darcy number, and the internal heat generation parameter on buoyancy-induced flow and heat transfer. The finite volume method with the power law scheme for convection terms is used to discretize the governing equations for momentum and energy, which are solved by the Gauss-Seidel and successive-over-relaxation methods. A systematic investigation on transient, steady fluid flow, and heat transfer phenomena under different physical conditions is carried out. The results obtained in the steady state regime are presented in the form of streamlines, isotherms, and midheight velocity profiles for various values of Grashof number, porosity, and Darcy number. It is found that the effect of density maximum is to slow down the buoyancy-driven convection and reduce the average heat transfer. The strength of convection and the heat transfer rate become weak due to more flow restriction in the porous medium for small porosity and high internal heat generation parameter.

Options are nowadays transacted within a lot of stock exchanges worldwide. The problem of options gets a special importance due to the fact that many of the managerial decisions can be assimilated to options. The paper deals with numerical methods of financial options evaluation. The mathematical model of determining the values of an option of European type is partial differential equation with an initial condition and two boundaries conditions, and for its solving, finite differences methods can be used. Out of these methods, the Crank-Nicolson method is proposed to be used. As Crank-Nicolson method is an implicit one, applying it leads to a linear system of equations, for whose solving the LU algorithm, Gauss method, Jacobi method, Gauss-Seidel method, method of successive over-relaxations are used.

Natural convection heat transfer in a square cavity induced by heated plate is studied numerically. Top and bottom of the cavity are adiabatic, the two vertical walls of the cavity have constant temperature lower than the plate’s temperature. The flow is assumed to be two-dimensional. The discretized equations were solved by finite difference method using Alternating Direction Implicit technique and Successive OverRelaxation method. The study was performed for different values of Grashof number ranging from 103 to 105 for different aspect ratios and position of heated plate. Air was chosen as a working fluid (Pr = 0.71). The effect of the position and aspect ratio of heated plate on heat transfer and flow were addressed. With increase of Gr heat transfer rate increased in both vertical and horizontal position of the plate. When aspect ratio of heated thin plate is decreased the heat transfer also decreases. For the vertical situation of thin plate heat transfer becomes more enhanced...

This paper is concerned with the generalized accelerated overrelaxation (GAOR) method, which constitutes a generalization of the basic iterative methods for the solution of linear systems. A number of new theoretical results are presented conceming the convergence theory of the GAOR method and special cases of it. Much attention is given to linear systems with positive definite coefficient matrices. Although the problem of determining the various optimum parameters of the GAOR method in the general case is very difficult to solve theoretically and is still an open one, the numerical examples which are presented in this paper show that a suitable exploitation of it may give much better results than the other basic iterative methods.

An intertemporal, spatial price equilibrium is determined for multiple commodities where the net import of each commodity in a given time period is a linear function of the prices of all commodities in that region and time period. Temporal and spatial flows are subject to fixed unit costs, and quotas in the form of upper bounds. A parallel decomposition scheme exploits characteristics of equilibria.

A spatial price equih~orium problem is modeled which allows piecewise linear convex flow costs, and a capacity limit on the trade flow between each supply/ demand pair of regions. Alternatively, the model determines the locations of intemaediate distr~ution centers in a matket economy composed of separate regions, each with approximately linear supply and demand functions. Equilibrium prices, regional supply and demand quantities, and commodity flows ate determined endogenously. The model has a quadratic programming formulation which is then reduced by exploiting the structure. The reduced model is particularly well suited to solution using successive over-relaxation. Keywords and phrases Spatial price equih'brium, quadratic programming, interregional trade, distribution, successive over-relaxation.

Tungsten inert Gas (TIG) welding takes place in an atmosphere of inert gas and uses a tungsten electrode. In this process heat input identification is a complex task and represents an important role in the optimization of the welding process. The technique used to estimate the heat flux is based on solution of an inverse three-dimensional transient heat conduction model with moving heat sources. The thermal fields at any region of the plate or at any instant are determined from the estimation of the heat rate delivered to the workpiece. The direct problem is solved by an implicit finite difference method. The system of linear algebraic equations is solved by Successive Over Relaxation method (SOR) and the inverse problem is solved using the Golden Section technique. The golden section technique minimizes an error square function based on the difference of theoretical and experimental temperature. The temperature measurements are obtained using thermocouples at accessible regions of the workpiece surface while the theoretical temperatures are calculated from the 3D transient thermal model.

In large-scale problems, classical Newton's method requires solving a large linear system of equations resulting from determining the Newton direction. This process often related as a very complicated process, and it requires a lot of computation (either in time calculation or memory requirement per iteration). Thus to avoid this problem, we proposed an improved way to calculate the Newton direction using an Accelerated Overrelaxation (AOR) point iterative method with two different parameters. To check the performance of our proposed Newton's direction, we used the Newton method with AOR iteration for solving unconstrained optimization problems with its Hessian is in arrowhead form and compared it with a combination of the Newton method with Gauss-Seidel (GS) iteration and the Newton method with Successive Over Relaxation (SOR) iteration. Finally, comparison results show that our proposed technique is significantly more efficient and more reliable than reference methods.

Tungsten inert Gas (TIG) welding takes place in an atmosphere of inert gas and uses a tungsten electrode. In this process heat input identification is a complex task and represents an important role in the optimization of the welding process. The technique used to estimate the heat flux is based on solution of an inverse three-dimensional transient heat conduction model with moving heat sources. The thermal fields at any region of the plate or at any instant are determined from the estimation of the heat rate delivered to the workpiece. The direct problem is solved by an implicit finite difference method. The system of linear algebraic equations is solved by Successive Over Relaxation method (SOR) and the inverse problem is solved using the Golden Section technique. The golden section technique minimizes an error square function based on the difference of theoretical and experimental temperature. The temperature measurements are obtained using thermocouples at accessible regions of the workpiece surface while the theoretical temperatures are calculated from the 3D transient thermal model.

A laboratory-experimental and theoretical-modeling investigation was conducted of isobaric, radiative cooling of cloud-like water mists to a remote heat sink, similar to what can happen at the tops of clouds. For mist initially at 20°C cooled by a radiative sink at −20°C, the mean (D43) mist droplet diameter grew from 5.5 to 8.4 μm and the mist temperature decreased from 20° to 3°C in just 80 s. Modeling showed that conventional assumptions were able to predict the measured temperature decrease reasonably well but not droplet size changes, suggesting that bulk radiative cooling was being reasonably well modeled but not detailed, droplet-size-dependent behavior. In a theoretical analysis, Lewis-number near unity was exploited to obtain an analytic expression for quasi-steady supersaturation that agrees with what Davies reported in 1985 but is simpler and is a function of only droplet size distribution, surface tension, and solute parameters and not radiative transfer. A simpler expre...

The numerical solutioit of vector field problems on digital computers is a slow process, especially when the characteristics of the regions investigated vary considerably. A method for the acceleration of the convergence of iterative procedures is described and applied to linear and non-linear problems. Physical considerations based on Stokes' theorem are utilized to modify Southwell's relaxation technique.

In this paper, Padé approximations are applied Black-Scholes model which reduces to heat equation. This paper shows various Padé approximaitons to obtain an e¤ective and accurate solution to the Black-Scholes equation for a European put/call option pricing problem. At the end of the paper, results of closed-form solution of Black-Scholes problem , solution of Crank-Nicolson approach and the solution of (1; 1), (1; 2), (2; 0), (2; 1), (2; 2) Padé approximations are given at a table.

The fundamental problem of linear algebra is to solve the system of linear equations (SOLE's). To solve SOLE's, is one of the most crucial topics in iterative methods. The SOLE's occurs throughout the natural sciences, social sciences, engineering, medicine and business. For the most part, iterative methods are used for solving sparse SOLE's. In this research, an improved iterative scheme namely, ''a new improved classical iterative algorithm (NICA)'' has been developed. The proposed iterative method is valid when the coefficient matrix of SOLE's is strictly diagonally dominant (SDD), irreducibly diagonally dominant (IDD), M-matrix, Symmetric positive definite with some conditions and H-matrix. Such types of SOLE's does arise usually from ordinary differential equations (ODE's) and partial differential equations (PDE's). The proposed method reduces the number of iterations, decreases spectral radius and increases the rate of convergence. Some numerical examples are utilized to demonstrate the effectiveness of NICA over Jacobi (J), Gauss Siedel (GS), Successive Over Relaxation (SOR), Refinement of Jacobi (RJ), Second Refinement of Jacobi (SRJ), Generalized Jacobi (GJ) and Refinement of Generalized Jacobi (RGJ) methods.

In the paper a description of heat transfer in a one-dimensional metal films is considered. The Boltzmann transport equation and a two-temperature model supplemented by appropriate boundary and initial conditions are applied to analyze the thermal process in a heated thin metal film. The problem considered is solved by the lattice Boltzmann method and the finite difference method respectively. The internal heat source is given in two different ways. In the first way, the internal heat source is considered a constant value, while in the second way an exponential function which simulates irradiation using a laser pulse is used. In the final part of the paper numerical examples of comparison of two methods and conclusions are presented.

The transient evolution of the flow and temperature fields inside two-dimensional cavities with inwards oscillating wall is discussed in the present work. The existence of openings in the cavity wall is considered. The flow governing equations are solved using a Vorticity-Stream Function formulation. Appropriate stream-function boundary conditions are used to describe the openings along the upper wall and the oscillation wall movement. Vorticity values along the cavity solids boundaries and the openings are solved by an iterative solution method. Density variations are considered by the use of Boussinesq approximation. Initially, analytical transformations are used to obtain a stationary solution domain and to cluster points near the upper wall. The transformed governing equations are discretized using an implicit Finite-Difference scheme. The resultant algebraic system is solved by an iterative solution method with sub-relaxation and local error control. The parametric study is pre...

In this paper, we propose an efficient parallel dynamic linear solver, called GPU-GMRES, for transient analysis of large power grid networks. The new method is based on the preconditioned generalized minimum residual (GMRES) iterative method implemented on heterogeneous CPU-GPU platforms. The new solver is very robust and can be applied to power grids with different structures and other applications like thermal analysis. The proposed GPU-GMRES solver adopts the very general and robust incomplete LU (ILU) based preconditioner. We show that by properly selecting the right amount of fill-ins in the incomplete LU factors, a good trade-off between GPU efficiency and GMRES convergence rate can be achieved for the best overall performance. Such a tunable feature makes this algorithm very adaptive to different problems. Furthermore, we properly partition the major computing tasks in GMRES solver to minimize the data traffic between CPU and GPU, which further boosts performance of the proposed method. Experimental results on the set of published IBM benchmark circuits and mesh-structured power grid networks show that the GPU-GMRES solver can deliver order of magnitudes speedup over the direct LU solver UMFPACK. GPU-GMRES can also deliver 3-10× speedup over the CPU implementation of the same GMRES method on transient analysis.

The zero-forcing (ZF) and minimum mean square error (MMSE) based detectors can approach optimal performance in the uplink of massive multiple-input multiple-output (MIMO) systems. However, they require inverting a matrix whose complexity is cubic in relation to the matrix dimension. This can lead to the high computational effort, especially in massive MIMO systems. To mitigate this, several iterative methods have been proposed in the literature. In this paper, we consider accelerated Chebyshev SOR (AC-SOR) and accelerated Chebyshev AOR (AC-AOR) algorithms, which improve the detection performance of conventional Successive Over-Relaxation (SOR) and Accelerated Over-Relaxation (AOR) methods, respectively. Additionally, we propose using a deep unfolding network (DUN) to optimize the parameters of the iterative AC-SOR and AC-AOR algorithms, leading to the AC-AORNet and AC-SORNet methods, respectively. The proposed DUN-based method leads to significant performance improvements compared to conventional iterative detectors for various massive MIMO channels. The results demonstrate that the AC-AORNet and AC-SORNet are effective, outperforming other state-of-theart algorithms. Furthermore, they are highly effective, particularly for high-order modulations such as 256-QAM (Quadrature Amplitude Modulation). Moreover, the proposed AC-AORNet and AC-SORNet require almost the same number of computations as AC-AOR and AC-SOR methods, respectively, since the use of deep unfolding has a negligible impact on the system's detection complexity. Furthermore, the proposed DUN features a fast and stable training scheme due to its smaller number of trainable parameters.

In order to solve the partial differential equations that arise in the Hartree-Fock theory for diatomic molecules and in molecular theories that include electron correlation, one needs efficient methods for solving partial differential equations. In this article, we present numerical results for a two-variable model problem of the kind that arises when one solves the Hartree-Fock equations for a diatomic molecule. We compare results obtained using the spline collocation and domain decomposition methods with third-order Hermite splines to results obtained using the more-established finite difference approximation and the successive over-relaxation method. The theory of domain decomposition presented earlier is extended to treat regions that are divided into an arbitrary number of subregions by families of lines parallel to the two coordinate axes. While the domain decomposition method and the finite difference approach both yield results at the micro-Hartree level, the finite difference approach with a 9point difference formula produces the same level of accuracy with fewer points. The domain decomposition method has the strength that it can be applied to problems with a large number of grid points. The time required to solve a partial differential equation for a fine grid with a large number of points goes down as the number of partitions increases. The reason for this is that the length of time necessary for solving a set of linear equations in each subregion is very much dependent upon the number of equations. Even though a finer partition of the region has more subregions, the time for solving the set of linear equations in each subregion is very much smaller. This feature of the theory may well prove to be a decisive factor for solving the two-electron pair equation, which-for a diatomic molecule-involves solving partial differential equations with five independent variables. The domain decomposition theory also makes it possible to study complex molecules by dividing them into smaller fragments that

In this paper, an efficient and reliable algorithm has been established to solve the second kind of FIE based on the lower-order piecewise polynomial and the lower-order quadrature method, namely Half-sweep Composite Trapezoidal (HSCT), which was used to discretize any integral term. Furthermore, due to the benefit of the complexity reduction technique via the half-sweep iteration concept presented from previous studies based on the cell-centered approach, this paper attempts to derive an HSCT piecewise linear collocation approximation equation generated from the discretization process of the proposed problem by considering the distribution of node points with vertex-centered type. Using half-sweep collocation node points over the linear collocation approximation equation, we could construct a system of HSCT linear collocation approximation equations, whose coefficientmatrix is huge-scale and dense. Furthermore, to attain the piecewise linear collocation solution of this linear system, we considered the efficient algorithm of the Half-Sweep Successive Over-Relaxation (HSSOR) iterative method. Therefore, several numerical experiments of the proposed iterative methods have been implemented by solving three tested examples, and the obtained results that were based on three parameters, namely iteration quantity, accomplished time, and maximum absolute error, were recorded and compared against other two iterations, namely Full-Sweep Gauss-Seidel (FSGS) and Half-Sweep Gauss-Seidel (HSGS).

In the article, a differential scheme is created for the the first-order diffusion equation using the Crank-Nicolson method. The stability of the differential scheme was checked using the Neumann method. To solve the problem numerically, stability intervals were found using the Neman method. This work presents an analysis of the stability of the Crank-Nicolson scheme for the two-dimensional diffusion equation using Von Neumann stability analysis. The Crank-Nicolson scheme is a widely used numerical method for solving partial differential equations that combines the explicit and implicit schemes. The stability analysis is an important factor to consider when choosing a numerical method for solving partial differential equations, as numerical instability can cause inaccurate solutions. We show that the Crank-Nicolson scheme is unconditionally stable, meaning that it can be used for a wide range of parameters without being affected by numerical instability. Overall, the analysis and im...

This paper asks what factors influence the dissipation time of altocumulus clouds. The question is addressed using three-dimensional, large-eddy simulations of a thin, midlevel cloud that was observed by aircraft. The cloud might be aptly described as ''altostratocumulus'' because it was overcast and contained radiatively driven turbulence. The simulations are used to construct a budget equation of cloud water. This equation allows one to directly compare the four processes that diminish liquid: diffusional growth of ice crystals, large-scale subsidence, radiative heating, and turbulent mixing of dry air into the cloud. Various sensitivity studies are used to find the ''equivalent sensitivity'' of cloud decay time to changes in various parameters. A change from no sunlight to direct overhead sunlight decreases the lifetime of our simulated cloud as much as increasing subsidence by 1.2 cm s À1 , increasing ice number concentration by 780 m À3 , or decreasing above-cloud total water mixing ratio by 0.60 g kg À1. Finally, interactions among the terms in the cloud water budget are summarized in a ''budget term feedback matrix.'' It is able to diagnose, for instance, that in our particular simulations, the diffusional growth of ice is a negative feedback.

The fundamental problem of linear algebra is to solve the system of linear equations (SOLE's). To solve SOLE's, is one of the most crucial topics in iterative methods. The SOLE's occurs throughout the natural sciences, social sciences, engineering, medicine and business. For the most part, iterative methods are used for solving sparse SOLE's. In this research, an improved iterative scheme namely, ''a new improved classical iterative algorithm (NICA)'' has been developed. The proposed iterative method is valid when the coefficient matrix of SOLE's is strictly diagonally dominant (SDD), irreducibly diagonally dominant (IDD), M-matrix, Symmetric positive definite with some conditions and H-matrix. Such types of SOLE's does arise usually from ordinary differential equations (ODE's) and partial differential equations (PDE's). The proposed method reduces the number of iterations, decreases spectral radius and increases the rate of convergence. Some numerical examples are utilized to demonstrate the effectiveness of NICA over Jacobi (J), Gauss Siedel (GS), Successive Over Relaxation (SOR), Refinement of Jacobi (RJ), Second Refinement of Jacobi (SRJ), Generalized Jacobi (GJ) and Refinement of Generalized Jacobi (RGJ) methods.

In order to understand the effects of the landfill operation on groundwater flow behavior, 2D horizontal groundwater simulation model was carried out. The model saved the memory of computer and time consumption comparing with 3D groundwater flow model. However, the most difficulty is the assignment of collecting pipe boundary in the study site. Therefore, 2D vertical model was applied to calculate the change of groundwater table above the collecting pipe. This paper paid attention to examine the validation of the assignment of the collecting pipe boundary by applying the results of 2D vertical model. 2D horizontal model was coupled with the recharge model to solve the partial differential equation of groundwater flow. Finite difference method and iterative successive over relaxation were applied. The drainage volume of leachate collection was summed up in the whole landfill site and compared with the average volume of treated waste water. The study demonstrated that the results of 2D vertical model validated and can be applied to 2D horizontal groundwater flow simulation.

This paper presents a parallel algorithm for the finite-volume discretisation of the Poisson equation on three-dimensional arbitrary geometries. The proposed method is formulated by using a 2D horizontal block domain decomposition and interprocessor data communication techniques with message passing interface. The horizontal unstructured-grid cells are reordered according to the neighbouring relations and decomposed into blocks using a load-balanced distribution to give all processors an equal amount of elements. In this algorithm, two parallel successive over-relaxation methods are presented: a multi-colour ordering technique for unstructured grids based on distributed memory and a block method using reordering index following similar ideas of the partitioning for structured grids. In all cases, the parallel algorithms are implemented with a combination of an acceleration iterative solver. This solver is based on a parabolic-diffusion equation introduced to obtain faster solutions of the linear systems arising from the discretisation. Numerical results are given to evaluate the performances of the methods showing speedups better than linear.

This paper focuses on the implementation of sequential algorithms for the simulation of parabolic equation in solving the thermal control systems. The platform of the temperature behaviour prediction is based on printed circuit board. The numerical finite-difference method (FDM) is used to design the discretization of these partial differential equations. The results of finite-difference approximation are presented graphically. Numerical methods that are used in solving the problem are the method of Jacobi, Gauss-Seidel, Red-Black Gauss-Seidel, Successive Over Relaxation (SOR) and Red-Black Successive Over Relaxation. The numerical analysis is treated done in terms of time execution, computation complexity, number of iterations, accuracy and convergence rate.

Transient and steady two-dimensional natural convection flow of micropolar fluid in a rectangular cavity heated from below with cold sidewalls has been studied numerically. The governing transient equations are solved using Alternate Direct Implicit method, together with Successive Over Relaxation method for stream function equation. The flow pattern and its effect on heat transfer and skin friction is studied for different values of Rayleigh number, Prandtl number, vortex viscosity parameter and cavity length. It was found that the number of vortex cells formed inside the cavity is a characteristic of the cavity length, the vortex viscosity parameter and Rayleigh number. The results are also compared with the case of Newtonian fluid. The heat transfer rate from heated surface, in the case of micropolar fluid is less than that of the Newtonian fluid under the same physical conditions.

The fundamental problem of linear algebra is to solve the system of linear equations (SOLE's). To solve SOLE's, is one of the most crucial topics in iterative methods. The SOLE's occurs throughout the natural sciences, social sciences, engineering, medicine and business. For the most part, iterative methods are used for solving sparse SOLE's. In this research, an improved iterative scheme namely, ''a new improved classical iterative algorithm (NICA)'' has been developed. The proposed iterative method is valid when the coefficient matrix of SOLE's is strictly diagonally dominant (SDD), irreducibly diagonally dominant (IDD), M-matrix, Symmetric positive definite with some conditions and H-matrix. Such types of SOLE's does arise usually from ordinary differential equations (ODE's) and partial differential equations (PDE's). The proposed method reduces the number of iterations, decreases spectral radius and increases the rate of convergence. Some numerical examples are utilized to demonstrate the effectiveness of NICA over Jacobi (J), Gauss Siedel (GS), Successive Over Relaxation (SOR), Refinement of Jacobi (RJ), Second Refinement of Jacobi (SRJ), Generalized Jacobi (GJ) and Refinement of Generalized Jacobi (RGJ) methods.

In this paper, we apply Modified Homotopy Analysis Method (MHAM) to find appropriate solutions to Zakharov-Kuznetsov equations, which are of utmost importance in applied and engineering sciences. The proposed modification is an elegant coupling of the Homotopy Analysis Method (HAM) and Taylor's series. Numerical results, coupled with graphical representation, explicitly reveal the complete reliability of the proposed algorithm.

Loss of machining efficiency, part repair, and replacement of mechanical components due to friction and wear is a recurring problem for performance industrial system applications. Recent studies on surface modification and micro-scale texturing have shown successful results in reducing friction and wear of lubricated surfaces. By acting as lubricant reservoirs and wear particle receptacles, micro-scale artificial surface textures positively influence lubrication regimes. In the present experimental study, a Stribeck curve is generated to compare the tribological properties of untextured and textured surfaces created by modulationassisted machining. Aluminum 6061-T6 disks are mated with high-speed steel pins on a pin-on-disk tribometer configuration for varying speed and texture depth and density. The results suggest that the textured surfaces produced by modulation-assisted machining accelerate the appearance of the elasto-hydrodynamic regime, while also reducing friction by 56% and wear by almost 90%. The enhanced friction and wear reduction were obtained under the lower speeds studied. In general, the disk with shallower dimples presented lower values of friction under the conditions studied. No major differences were found for textures with different dimple densities.

In the Black-Scholes-Merton model, as well as in more general stochastic models in finance, the price of an American option solves a parabolic variational inequality. When the variational inequality is discretized, one obtains a linear complementarity problem that must be solved at each time step. This paper presents an algorithm for the solution of these types of linear complementarity problems that is significantly faster than the methods currently used in practice. The new algorithm is a two-phase method that combines the active-set identification properties of the projected SOR iteration with the second-order acceleration of a (recursive) reduced-space phase. We show how to design the algorithm so that it exploits the structure of the linear complementarity problems arising in these financial applications and present numerical results that show the effectiveness of our approach.

We investigate numerical solution of finite difference approximations to American option pricing problems, using a new direct numerical method: simplex solution of a linear programming formulation. This approach is based on an extension to the parabolic case of the equivalence between linear order complementarity problems and abstract linear programs known for certain elliptic operators. We test this method empirically, comparing simplex and interior point algorithms with the projected successive overrelaxation (PSOR) algorithm applied to the American vanilla and lookback puts. We conclude that simplex is roughly comparable with projected SOR on average (faster for fine discretizations, slower for coarse), but is more desirable for robustness of solution time under changes in parameters. Furthermore, significant speedups over the results given here have been achieved and will be published elsewhere.

This study develops the novel fourth-order iterative alternating decomposition explicit (IADE) method of Mitchell and Fairweather (IADEMF4) algorithm for the solution of the one-dimensional linear heat equation with Dirichlet boundary conditions. The higher-order finite difference scheme is developed by representing the spatial derivative in the heat equation with the fourth-order finite difference Crank-Nicolson approximation. This leads to the formation of pentadiagonal matrices in the systems of linear equations. The algorithm also employs the higher accuracy of the Mitchell and Fairweather variant. Despite the scheme’s higher computational complexity, experimental results show that it is not only capable of enhancing the accuracy of the original corresponding method of second-order (IADEMF2), but its solutions are also in very much agreement with the exact solutions. Besides, it is unconditionally stable and has proven to be convergent. The IADEMF4 is also found to be more accur...

This paper presents the application of a half-sweep iteration concept to the Grünwald implicit difference schemes with the Kaudd Successive Over-Relaxation (KSOR) iterative method in solving one-dimensional linear time-fractional parabolic equations. The formulation and implementation of the proposed methods are discussed. In order to validate the performance of HSKSOR, comparisons are made with another two iterative methods, full-sweep KSOR (FSKSOR) and Gauss-Seidel (FSGS) iterative methods. Based on the numerical results of three tested examples, it shows that the HSKSOR is superior compared to FSKSOR and FSGS iterative methods.

In this article, we introduce an implicit finite difference approximation for one-dimensional porous medium equations using Quarter-Sweep approach. We approximate the solutions of the nonlinear porous medium equations by the application of the Newton method and use the Gauss-Seidel iteration. This yields a numerical method that reduces the computational complexity when the spatial grid spaces are reduced. The numerical result shows that the proposed method has a smaller number of iterations, a shorter computation time and a good accuracy compared to Newton-Gauss-Seidel and Half-Sweep Newton-Gauss-Seidel methods.

In this paper, we consider the application of the Newton the approximate solution of the two nonlinear finite difference approximation equation to implicit finite difference scheme. The developed nonlinear system is linearized by using the Newton method. At each temporal step, the corresponding linear systems are solved by using SOR iteration. We investigate the eff three examples of 2D PME and the performance is compared with the Newton method. Numerical results show that the Newton Newton-GS iterative method in terms of a number of iterations, computer time and maximum absolute errors.