Successive Overrelaxation Iterative Method Research Articles

Lippmann-Schwinger equation representation of Green's function and its preconditioned generalized over-relaxation iterative solution in wavelet domain

The calculation of Green's function is the core of seismic forward and inverse methods based on integral operators. When the Lippmann-Schwinger (L-S) equation is used to calculate Green's function in strongly scattering media, both the Born scattering series and the numerical iterative method encounter issues of slow convergence or divergence. Although the renormalization method derived from quantum mechanics can effectively address the convergence problem of Born scattering series in strong scattering problems, it is acknowledgeed that the convergence conditions and rates of convergence of different reformulation series may vary, and no universal convergence reformulation scattering series exists. Numerical methods for solving integral equations tend to be more general and mathematically robust. In this work, we focus on the numerical solution method of L-S equations. By using a wavelet-domain preconditioner to a reformulated or equivalent Lippmann-Schwinger (L-S) equation, we present an iterative method for numerically solving the equivalent L-S equation aimed at improving the rate of convergence and iteration efficiency in strongly inhomogeneous media. Following Jakobsen et al. (2020), we first introduce a small imaginary component into the background wave number，then rewrite the L-S equation to derive the equivalent complex wave number L-S equation. This reformulation ensures that the coefficient matrix exhibits a banded structure after numerical discretization, allowing the wavelet coefficient matrix to maintain good sparsity. We employ a multi-level fill-in incomplete LU (ILU) factorization method along with a block ILU-based algebraic recursive multilevel solve (ARMS) method in the wavelet domain to generate sparse approximate inverses as preconditioning operators, thereby accelerating the convergence of the generalized successive over-relaxation (GSOR) iterative method. This method is applied to compute numerical Green's functions in strongly inhomogeneous media. Numerical results demonstrate that our method yields simulation outcomes consistent with those obtained from the direct method for solving the original real wave number L-S equation. By testing various preconditioners, we find that the ARMS preconditioner offers significant advantages in operator generation efficiency and non-zero element filling ratio, effectively accelerating the convergence of the GSOR iterative method while achieving higher computational efficiency.

Explicit Group Iterative Method with Semi-Approximate Implicit Approach for Solving One-Dimensional Burgers’ Equation

This paper is considering a semi-approximate approach in investigating the Burgers’ problem. The process of the discretization of Burgers’ problem has taken place which it starts with the second-order finite difference in the discretize process together with the linearization part by using the semi-approximate implicit scheme in a way to achieve the approximation equation, thus generating the corresponding linear system equations. Besides, the Gauss-Seidel (GS), Successive Over-Relaxation (SOR) and explicit group (EG) iterative method has combined together with the SOR iterative method, namely as 4-point EGSOR (4EGSOR) has been introduced in this study for resolving the linear system. To assess the proficiency of the suggested methods on the approximation equation, the numerical test has been conducted by considering three parameters, which are computational time, iteration number and maximum absolute error. The findings indicate that the 4EGSOR method outperforms both SOR and GS iterative methods.

Numerical Optimization Analysis of Floating Ring Seal Performance Based on Surface Texture

Much research and practical experience have shown that the utilization of textures has an enhancing effect on the performance of dynamic seals and the dynamic pressure lubrication of gas bearings. In order to optimize the performance of floating ring seals, this study systematically analyzes the effects of different texture shapes and their parameters. The Reynolds equation of the gas is solved by the successive over-relaxation (SOR) iteration method. The pressure and thickness distributions of the seal gas film are solved to derive the floating force, end leakage, friction, and the ratio of buoyancy to leakage within the seal. The effects of various texture shapes, including square, 2:1 rectangle, triangle, hexagon, and circle, as well as their parameters, such as texture depth, angle, and area share, on the sealing performance are discussed. Results show that the texture can increase the air film buoyancy and reduce friction, but it also increases the leakage by a small amount. Square textures and rectangular textures are relatively effective. The deeper the depth of the texture within a certain range, the better the overall performance of the floating ring seal. As the texture area percentage increases, leakage tends to increase and friction tends to decrease. A fractal roughness model is developed, the effect of surface roughness on sealing performance is briefly discussed, and finally the effect of surface texture with roughness is analyzed. Some texture parameters that can significantly optimize the sealing performance are obtained. Rectangular textures with certain parameters enhance the buoyancy of the air film by 81.2%, which is the most significant enhancement effect. This rectangular texture reduces friction by 25.8% but increases leakage by 79.5%. The triangular textures increase buoyancy by 28.02% and leakage increases by only 10.08% when the rotation speed is 15,000 r/min. The results show that texture with appropriate roughness significantly optimizes the performance of the floating ring seal.

Computation of Green’s Function in a Strongly Heterogeneous Medium Using the Lippmann–Schwinger Equation: A Generalized Successive Over-Relaxion plus Preconditioning Scheme

The computation of Green’s function is a basic and time-consuming task in realizing seismic imaging using integral operators because the function is the kernel of the integral operators and because every image point functions as the source point of Green’s function. If the perturbation theory is used, the problem of the computation of Green’s function can be transformed into one of solving the Lippmann–Schwinger (L–S) equation. However, if the velocity model under consideration has large scale and strong heterogeneity, solving the L–S equation may become difficult because only numerical or successive approximate (iterative) methods can be used in this case. In the literature, one of these methods is the generalized successive over-relaxation (GSOR) iterative method, which can effectively solve the L–S equation and obtain the desired convergent iterative series. However, the GSOR iterative method may encounter slow convergence when calculating the high-frequency Green’s function. In this paper, we propose a new scheme that utilizes the GSOR iterative with a precondition to solve the complex wavenumber L–S equation in a slightly attenuated medium. The complex wavenumber with imaginary components localizes the energy of the background Green’s function and reduces its singularity by enabling exponential decay. Introduction of the preconditioning operator can further improve the convergence speed of the GSOR iterative series. Then, we provide a preconditioned generalized successive over-relaxation (pre-GSOR) iterative format. Our numerical results show that if an appropriate damping factor and a proper preconditioning operator are selected, the method presented here outperforms the GSOR iterative for the real wavenumber L–S equation in terms of the convergence speed, accuracy, and adaptation to high frequencies.

Comprehensive Analysis of Internal Flow Dynamics in Pipeline Systems

This research delves into the intricate flow dynamics within a gas pipeline, leveraging the small perturbation equations to elucidate the flow scenario. Utilizing the Python programming language, the study derives the numerical solutions for the governing equations of the pipeline's flow field. This is achieved through the successive over-relaxation (SOR) iterative method in tandem with the Neumann boundary conditions. The flow dynamics are subsequently visualized and analyzed using streamlined diagrams and Mach number cloud representations, facilitating the identification of stress concentration points on the pipeline's inner wall. Such insights significantly affect the pipeline's optimal design and strategic reinforcement. Beyond its immediate application, the methodology presented herein offers a robust framework for addressing diverse fluid mechanics simulation challenges. Its versatility extends to realms such as automotive contour design, pipeline leakage simulations, and engine compressor configurations. When confronted with the need to dissect fluid flow and stress dynamics, professionals can harness the Python-based SOR iteration code to derive governing fluid equations, enabling a graphical interpretation through streamlined and Mach number cloud diagrams. This study, thus, underscores a pivotal toolset for fluid mechanics challenges across various engineering domains.

Thermo-Hydrodynamic Lubrication Analysis of Slipper Pair Considering Wear Profile

The profile of sealing land is a sensitive factor affecting the thermo-hydrodynamic lubrication characteristics of the slipper pair. In this paper, the non-uniform wear of the running surface under the slipper was presented and defined as the boundary condition. Based on the finite volume method and the successive over-relaxation iteration method, a discrete numerical model coupled with the temperature, pressure, and thickness of the oil film was constructed. The Newton and sequential circulation methods were used to solve the coupling equations. The influence of the wear profile on the film thickness, sliding attitude, and leakage were discussed. The analyzed results show that the control of the wear on the outer side of sealing land and the contour vertex position, and the avoidance of the wear on the inner side of sealing land could improve the thermo-hydrodynamic lubrication performance of the slipper pair.

Preconditioned successive over relaxation iterative method via semi-approximate approach for Burgers’ equation

<span lang="EN-US">This paper proposes the combination of a preconditioner applied with successive over relaxation (SOR) iterative method for solving a sparse and huge scale linear system (LS) in which its coefficient matrix is a tridiagonal matrix. The purpose for applying the preconditioner is to enhance the convergence rate of SOR iterative method. Hence, in order to examine the feasibility of the proposed iterative method which is preconditioner SOR (PSOR) iterative method, first we need to derive the approximation equation of one-dimensional (1D) Burgers’ equation through the discretization process in which the second-order implicit finite difference (SIFD) scheme together with semi-approximate (SA) approach have been applied to the proposed problem. Then, the generated LS is modified into preconditioned linear system (PLS) to construct the formulation of PSOR iterative method. Furthemore, to analyze the feasibility of PSOR iterative method compared with other point iterative methods, three examples of 1D Burgers’ equation are considered. In conclusion, the PSOR iterative method is superior than PGS iterative method. The simulation results showed that our proposed iterative method has low iteration numbers and execution time.</span>

Effect of the phase-transition fluid reaction heat on wellbore temperature in self-propping phase-transition fracturing technology

The key to the success of self-propping phase-transition fracturing (SPF) technology using two immiscible fluids to generate proppants in-situ in a reservoir lies in accurate calculations of temperature distribution. As the reaction heat of phase-transition fluid (PF) significantly affects wellbore temperature, reaction kinetic parameters were fitted by experimental data based on the Arrhenius equation, and the transient temperature model considering the reaction heat is established based on the first law of thermodynamics. This model is discretized by the finite difference method and solved by the successive over-relaxation iteration method. The results show that the reaction heat effect on wellbore temperature cannot be ignored. A temperature value and a phase transition time at the well bottom are the largest in the whole wellbore, so the phase transition ratio at the well bottom is the largest. Moreover, since the PF with incomplete phase transition in a wellbore is easier to enter fractures and prop fracture fronts, it is recommended to inject a pre-pad fracturing fluid before injecting PF to reduce wellbore temperature and prevent premature phase transition in the wellbore. These findings can help reveal the action mechanisms of different injection methods and parameters in a heat transfer process, which is of great significance for the theoretical research and field implementation of SPF technology.

Control Model and Optimization Study of Temperature Distribution Applied in Thermite Plugging and Abandonment Technology

Abstract With the continuous exploration and development of oil wells, we must pay attention to the risk of leakage from abandoned wells. Therefore, it is necessary to plug and abandon the abandoned well. However, there are many limitations in the traditional plugging and abandonment (P&A) operation, for example, cement's bearing capacity, the cement's corrosion resistance, the problem of the extended operation time, and high cost. To overcome the aforementioned issues, a thermite plugging and abandonment (TP&A) technology is proposed in this field. The technology uses the aluminothermic reaction to melt the original or set materials for P&A operation. To promote the phase transformation of more materials in the well to form a plug with good plugging performance, the temperature distribution in the TP&A system was optimized. Based on the heat conduction theory and successive overrelaxation iterative method, a heat conduction model based on the temperature release law of aluminothermic reaction is established and solved. The temperature change law under different combinations of the downhole environment is studied. The optimized model can maintain the high-efficiency transfer of energy, fluid–structure interaction, and the interaction between fluids. The material after the phase change can be cooled to form a plug with good plugging performance.

Numerical Modeling of Fracture Height Propagation in Multilayer Formations Considering the Plastic Zone and Induced Stress.

Hydraulic fracturing and acid fracturing are very effective stimulation technologies and are widely used in unconventional reservoir development. Fracture height, as an essential parameter to describe the geometric size of a fracture, is not only the input parameter of two-dimensional fracturing models but also the output parameter of three-dimensional fracturing models. Accurate prediction of fracture height growth can effectively avoid some risks. For example, petroleum reservoirs produce a large amount of formation water because wrong fracture height prediction leads to the connection between the oil or gas reservoir and the water layer. Although some fracture height prediction models were developed, few models considered the effects of the plastic zone, induced stress, and heterogeneous multilayer formation and its interaction. Therefore, considering the influence of many factors, an improved fracture-equilibrium-height model was developed in this study. The successive over-relaxation iteration method and the displacement discontinuity method were used to solve the model. We investigated the effects of the geological and engineering factors on fracture height growth by using the model, and some important conclusions were obtained. The higher the fracture height, the larger the plastic zone size, and the more obvious its influence on fracture height propagation. High overlying or underlying in situ stress and fracture toughness and low fluid density played a positive role in limiting the growth of the fracture height. Induced stress caused by fracture 1 could not only inhibit the height growth of fracture 2 but also promote its growth. The model established in this paper could be coupled to a fracturing simulator to provide a more reliable fracture height prediction.

Half-Sweep Refinement of SOR Iterative Method via Linear Rational Finite Difference Approximation for Second-Order Linear Fredholm Integro-Differential Equations

The numerical solutions of the second-order linear Fredholm integro-differential equations have been considered and discussed based on several discretization schemes. In this paper, the new schemes are developed derived on the hybrid of the three-point half-sweep linear rational finite difference (3HSLRFD) approaches with the half-sweep composite trapezoidal (HSCT) approach. The main advantage of the established schemes is that they discretize the differential terms and integral term of second-order linear Fredholm integro-differential equations into the algebraic equations and generate the corresponding linear system. Furthermore, the half-sweep (HS) concept is combined with the refinement of the successive over-relaxation (RSOR) iterative method to create the new half-sweep successive over-relaxation (HSRSOR) iterative method, which is implemented to get the numerical solution of a system of linear algebraic equations. Apart from that, the classical or full-sweep Gauss-Seidel (FSGS) and full-sweep successive over-relaxation iterative (FSSOR) methods are presented, which serve as the control method in this paper. In the end, we employed FSGS, FSRSOR and HSRSOR methods to obtain numerical solutions of three examples and make a detailed comparison from three aspects of the number of iterations, elapsed time and maximum absolute error. Numerical results demonstrate that FSRSOR and HSRSOR methods have lesser iterations, faster elapsed time, and are more accurate than FSGS. In addition, HSRSOR is the most effective of the three methods. To sum up, this paper has successfully proposed the applicability and superiority of the new HSRSOR method based on 3HSLRFD-HSCT schemes.

Refinement of SOR Iterative Method for the Linear Rational Finite Difference Solution of Second-Order Fredholm Integro-Differential Equations

The primary objective of this paper is to develop the Refinement of Successive Over-Relaxation (RSOR) method based on a three-point linear rational finite difference-quadrature discretization scheme for the numerical solution of second-order linear Fredholm integro-differential equation (FIDE). Besides, to illuminate the superior performance of the proposed method, some numerical examples are presented and solved by implementing three approaches which are the Gauss-Seidel (GS), the Successive Over-Relaxation (SOR) and the RSOR methods. Lastly, through the comparison of the results, it is verified that the RSOR method is more effective than the other two methods, especially when considering the aspects of the number of iterations and running time.

Modulus-based Successive Overrelaxation Iteration Method for Pricing American Options with the Two-asset Black–Scholes and Heston's Models Based on Finite Volume Discretization

In this paper we introduce a new numerical method for the linear complementarity problems (LCPs) arising from two-asset Black–Scholes and Heston's stochastic volatility American options pricing. Based on barycenter dual mesh, a class of finite volume method (FVM) is proposed for the spatial discretization, coupled with the backward Euler and Crank–Nicolson schemes are employed for time stepping of the partial differential equations (PDEs). Then, for the resulting time-dependent LCPs are solved by using an efficient modulus-based successive overrelaxation (MSOR) iteration method. Numerical experiments are carried out to verify the efficiency and usefulness of the proposed method.

MN-PGSOR Method for Solving Nonlinear Systems with Block Two-by-Two Complex Symmetric Jacobian Matrices

For solving the large sparse linear systems with 2 × 2 block structure, the generalized successive overrelaxation (GSOR) iteration method is an efficient iteration method. Based on the GSOR method, the PGSOR method introduces a preconditioned matrix with a new parameter for the coefficient matrix which can enhance the efficiency. To solve the nonlinear systems in which the Jacobian matrices are complex and symmetric with the block two-by-two form, we try to use the PGSOR method as an inner iteration, with the help of the modified Newton method as an efficient outer iteration method. This new method is called the modified Newton-PGSOR (MN-PGSOR) method. Local convergence properties of the MN-PGSOR are analyzed under the Hölder condition. Finally, we give the comparison of our new method with some previous methods in the numerical results. The MN-PGSOR method is superior in both iteration steps and computing time.

QSSOR and cubic non-polynomial spline method for the solution of two-point boundary value problems

Two-point boundary value problems are commonly used as a numerical test in developing an efficient numerical method. Several researchers studied the application of a cubic non-polynomial spline method to solve the two-point boundary value problems. A preliminary study found that a cubic non-polynomial spline method is better than a standard finite difference method in terms of the accuracy of the solution. Therefore, this paper aims to examine the performance of a cubic non-polynomial spline method through the combination with the full-, half-, and quarter-sweep iterations. The performance was evaluated in terms of the number of iterations, the execution time and the maximum absolute error by varying the iterations from full-, half- to quarter-sweep. A successive over-relaxation iterative method was implemented to solve the large and sparse linear system. The numerical result showed that the newly derived QSSOR method, based on a cubic non-polynomial spline, performed better than the tested FSSOR and HSSOR methods.

Numerical Analysis Based on Oil Film Characteristics of Sliding Bearing

Taking the finite-length sliding bearing as the research object, the Reynolds equation is solved by the finite difference method and the successive over-relaxation iteration method to obtain the oil film pressure. It is concluded that the dimensionless oil film characteristics are related to bearing eccentricity and length diameter ratio. In order to study the oil film characteristics of sliding bearing, the bearing eccentricity and length diameter ratio are taken as the research objects, and the oil film thickness, load component, attitude angle and end discharge flow are solved by numerical analysis. It is concluded that with the increase of eccentricity, the bearing capacity of oil film increases, the attitude angle decreases and the end discharge flow increases with the case of a certain length diameter ratio.

On the modulus-based successive overrelaxation iteration method for horizontal linear complementarity problems arising from hydrodynamic lubrication

In this paper, for the horizontal linear complementarity problems arising from hydrodynamic lubrication, the convergence of modulus-based successive overrelaxation iteration method is presented, which extends the existing results given in the recent work of Mezzadri F. and Galligani E. published on Math. Comput. Simulat. Numerical examples demonstrate the significance of the proposed theorems.

Numerical Evaluation of Quarter-Sweep KSOR Method to Solve Time-Fractional Parabolic Equations

We study the performance of the combination of quarter-sweep iteration concept with the Kaudd Successive Over-Relaxation (KSOR) iterative method in solving the discretized one-dimensional time-fractional parabolic equation. We called the mixed of these two concepts as QSKSOR. The time-fractional derivative in Grunwald sense, together with the implicit finite difference scheme was used to discretized the tested problems to form the quarter-sweep implicit finite difference approximation equations in the sense of Grunwald type. This approximation equation of half-sweep will then generate a linear system. Next, we used the proposed QSKSOR iterative method to the generated linear systems before comparing the effectiveness between the other family of KSOR method, FSKSOR and HSKSOR with respect to the full- and half-sweep cases respectively. To do so, three examples are included. The results of this study show the superiority of the QSKSOR iterative method in terms of iteration numbers and execution time in comparison to the other two methods.

Modified Newton-GSOR method for solving complex nonlinear systems with symmetric Jacobian matrices

In this paper, a new method for solving complex nonlinear systems with symmetric Jacobian matrices is proposed—modified Newton generalized successive overrelaxation (MN-GSOR) iteration method. The new MN-GSOR method helps us to solve nonlinear systems using the GSOR method as an inner iterative approximation for solving the Newton equations. Next, we use the Holder continuous condition instead of the Lipschitz assumption to analyze and prove the convergence properties of the MN-GSOR method. Numerical experiments are conducted to show the superiority and effectiveness of MN-GSOR method, and comparisons are made to see the advantages over some other recently proposed methods. Furthermore, when the imaginary part of the Jacobian matrix is symmetric but non-definite, some recently proposed methods are not applicable, but the MN-GSOR method still works well.

Grünwald Implicit Solution of One-Dimensional Time-Fractional Parabolic Equations Using HSKSOR Iteration

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.