Sixth-order Convergence Research Articles

A Sixth-Order Cubic B-Spline Approach for Solving Linear Boundary Value Problems: An In-Depth Analysis and Comparative Study

This research presents an efficient and highly accurate cubic B-spline method (CBSM) for solving second-order linear boundary value problems (BVPs). The method achieves sixth-order convergence, supported by rigorous error analysis, ensuring rapid error reduction with mesh refinement. The effectiveness of the CBSM is validated through four numerical examples, showcasing its accuracy, reliability, and computational efficiency, making it well-suited for large-scale problems. A comparative analysis with existing methods confirms the superior performance of the CBSM, positioning it as a practical and powerful tool for solving second-order BVPs.

Generalized high-order iterative methods for solutions of nonlinear systems and their applications

<abstract><p>In this paper, we have constructed a family of three-step methods with sixth-order convergence and a novel approach to enhance the convergence order $ p $ of iterative methods for systems of nonlinear equations. Additionally, we propose a three-step scheme with convergence order $ p+3 $ (for $ p\geq3 $) and have extended it to a generalized $ (m+2) $-step scheme by merely incorporating one additional function evaluation, thus achieving convergence orders up to $ p+3m $, $ m\in\mathbb{N} $. We also provide a thorough local convergence analysis in Banach spaces, including the convergence radius and uniqueness results, under the assumption of a Lipschitz-continuous Fréchet derivative. Theoretical findings have been validated through numerical experiments. Lastly, the performance of these methods is showcased through the analysis of their basins of attraction and their application to systems of nonlinear equations.</p></abstract>

Sixth-order hybrid finite difference methods for elliptic interface problems with mixed boundary conditions

In this paper, we develop sixth-order hybrid finite difference methods (FDMs) for the elliptic interface problem −∇⋅(a∇u)=f in Ω﹨Γ, where Γ is a smooth interface inside Ω. The variable scalar coefficient a>0 and source f are possibly discontinuous across Γ. The hybrid FDMs utilize a 9-point compact stencil at any interior regular points of the grid and a 13-point stencil at irregular points near Γ. For interior regular points away from Γ, we obtain a sixth-order 9-point compact FDM satisfying the sign and sum conditions for ensuring the M-matrix property. We also derive sixth-order compact (4-point for corners and 6-point for edges) FDMs satisfying the sign and sum conditions for the M-matrix property at any boundary point subject to (mixed) Dirichlet/Neumann/Robin boundary conditions. Thus, for the elliptic problem without interface (i.e., Γ is empty), our compact FDM has the M-matrix property for any mesh size h>0 and consequently, satisfies the discrete maximum principle, which guarantees the theoretical sixth-order convergence. For irregular points near Γ, we propose fifth-order 13-point FDMs, whose stencil coefficients can be effectively calculated by recursively solving several small linear systems. Theoretically, the proposed high order FDMs use high order (partial) derivatives of the coefficient a, the source term f, the interface curve Γ, the two jump functions along Γ, and the functions on ∂Ω. Numerically, we always use function values to approximate all required high order (partial) derivatives in our hybrid FDMs without losing accuracy. Our proposed FDMs are independent of the choice representing Γ and are also applicable if the jump conditions on Γ only depend on the geometry (e.g., curvature) of the curve Γ. Our numerical experiments confirm the sixth-order convergence in the l∞ norm of the proposed hybrid FDMs for the elliptic interface problem.

Nine-point compact sixth-order approximation for two-dimensional nonlinear elliptic partial differential equations: Application to bi- and tri-harmonic boundary value problems

Nine point sixth order compact numerical approximations are suggested to solve 2D nonlinear elliptic partial differential equations (NLEPDEs) and for the estimation of normal derivatives on a uniform rectangular grid subject to Dirichlet boundary conditions. We deliberate error analysis and reveal that, under specific conditions, our method converges to the sixth order. In addition, we extend our technique to vector form in order to solve the system of NLEPDEs. In application, we discuss nine-point compact sixth order approximations for bi- and tri-harmonic elliptic boundary value problems. Numerical experiments are carried out on several benchmark problems including bi- and tri-harmonic equations, and verified the sixth order convergence of the proposed methods.

Numerical method of sixth order convergence for solving a fourth order nonlinear boundary value problem

In this paper, we construct an iterative method of sixth order convergence for solving a fourth order nonlinear boundary value problem. The idea of the method is to construct difference schemes of sixth order accuracy for solving Dirichlet problems for second order equation at each iteration of the iterative method on continuous level constructed before by ourselves. Some numerical examples demonstrate the validity of obtained theoretical results.

A sixth order quasi-compact finite difference method for Helmholtz equations with variable wave numbers

A sixth order quasi-compact finite difference (SQC) scheme for the Helmholtz equation with variable wave numbers in two and three dimensional rectangular domains has been developed and analyzed. The finite difference coefficients for the solution and the weights for the source term have explicit expressions without involving derivatives of the source term. The proof of the sixth order convergence of the new method is also presented. Numerical experiments presented in this paper confirmed the order of convergence of the new method.

Optimized Quadrature Rules for Isogeometric Frequency Analysis of Wave Equations Using Cubic Splines

An optimization of quadrature rules is presented for the isogeometric frequency analysis of wave equations using cubic splines. In order to optimize the quadrature rules aiming at improving the frequency accuracy, a frequency error measure corresponding to arbitrary four-point quadrature rule is developed for the isogeometric formulation with cubic splines. Based upon this general frequency error measure, a superconvergent four-point quadrature rule is found for the cubic isogeometric formulation that achieves two additional orders of frequency accuracy in comparison with the sixth-order accuracy produced by the standard approach using four-point Gauss quadrature rule. One interesting observation is that the first and last integration points of the superconvergent four-point quadrature rule go beyond the domain of conventional integration element. However, these exterior integration points pose no difficulty on the numerical implementation. Subsequently, by recasting the general four-point quadrature rule into a three-point formation, the proposed frequency error measure also reveals that the three-point Gauss quadrature rule is unique among possible three-point rules to maintain the same sixth-order convergence rate as the four-point Gauss quadrature rule for the cubic isogeometric formulation. These theoretical results are clearly demonstrated by numerical examples.

Compact 9-point finite difference methods with high accuracy order and/or M-matrix property for elliptic cross-interface problems

In this paper we develop finite difference schemes for elliptic problems with piecewise continuous coefficients that have (possibly huge) jumps across fixed internal interfaces. In contrast with such problems involving one smooth non-intersecting interface, that have been extensively studied, there are very few papers addressing elliptic interface problems with intersecting interfaces of coefficient jumps. It is well known that if the values of the permeability in the four subregions around a point of intersection of two such internal interfaces are all different, the solution has a point singularity that significantly affects the accuracy of the approximation in the vicinity of the intersection point. In the present paper we propose a fourth-order 9-point finite difference scheme on uniform Cartesian meshes for an elliptic problem whose coefficient is piecewise constant in four rectangular subdomains of the overall two-dimensional rectangular domain. Moreover, for the special case when the intersecting point of the two lines of coefficient jumps is a grid point, such a compact scheme, involving relatively simple formulas for computation of the stencil coefficients, can even reach sixth order of accuracy. Furthermore, we show that the resulting linear system for the special case has an M-matrix, and prove the theoretical sixth order convergence rate using the discrete maximum principle. Our numerical experiments demonstrate the sixth (for the special case) and at least fourth (for the general case) accuracy orders of the proposed schemes. In the general case, we derive a compact third-order finite difference scheme, also yielding a linear system with an M-matrix. In addition, using the discrete maximum principle, we prove the third order convergence rate of the scheme for the general elliptic cross-interface problem.

Convergence and dynamical study of a new sixth order convergence iterative scheme for solving nonlinear systems

<abstract><p>A novel family of iterative schemes to estimate the solutions of nonlinear systems is presented. It is based on the Ermakov-Kalitkin procedure, which widens the set of converging initial estimations. This class is designed by means of a weight function technique, obtaining 6th-order convergence. The qualitative properties of the proposed class are analyzed by means of vectorial real dynamics. Using these tools, the most stable members of the family are selected, and also the chaotical elements are avoided. Some test vectorial functions are used in order to illustrate the performance and efficiency of the designed schemes.</p></abstract>

Extended comparison between two Newton–Jarratt sixth order schemes for nonlinear models under the same set of conditions

Two sixth order convergence order schemes are compared and extended to solve Banach space valued models. Earlier studies have used derivatives and Taylor expansions up to order seven to show the convergence order in a finite-dimensional Euclidean space se

Finding Approximate Roots of Nonlinear Equations by Using New Sixth-Order Convergence

In this study, we introduced new iterative techniques, for the purpose solving nonlinear equations. The new approaches are based on the auxiliary equation and Newton's method, respectively. Our method's a convergence analysis is explained. The novel approach is found to have convergence order of six. Numerical experiments show that the new method work better than well-known iterative methods and is similar to them.

ON THE CONVERGENCE OF THE SIXTH ORDER HOMEIER LIKE METHOD IN BANACH SPACES

A sixth order Homeier-like method is introduced for approximating a solution of the non-linear equation in Banach space. Assumptions only on first and second derivatives are used to obtain a sixth order convergence. Our proof does not depend on Taylor series expansions as in the earlier studies for the similar methods.

Reduced cost numerical methods of sixth-order convergence for systems of nonlinear models

Reduced cost numerical methods of sixth-order convergence for systems of nonlinear models

Sixth-order quasi-compact difference schemes for 2D and 3D Helmholtz equations

Sixth-order quasi-compact difference (QCD) schemes are proposed for the two-dimensional (2D) and the three-dimensional (3D) Helmholtz equations with the variable parameter. Our approach provides the compact mesh stencil for the unknowns, while the noncompact mesh stencil is employed for the source term and the parameter function without involving their derivatives. For the proper interior grid points that are without adjoining the boundary, the sixth-order truncated errors are obtained by the QCD method. Yet the compact scheme is utilized for both of the source term and the parameter function on the improper interior grids that neighbor the boundary, which only reaches the fourth-order local truncated errors. Theoretically, the sixth-order accuracy of the global error by the proposed QCD method is strictly proved for the non-positive constant parameter. Numerical examples are given to demonstrate that the QCD method achieves the global sixth-order convergence for general variable parameters.

A New Multi-Step Iterative Method for Solving Non-Linear Equations

The scope of this paper is to establish a new three-step sixth-order iterative method to find the root of non-linear equations. This paper came up with a further modification of Newton’s method with higher-order convergence and proved that this scheme is of the sixth order of convergence. Finally, we tested several problems to show the method’s efficiency over the existing ones.

A high‐order efficient technique and its convergence analysis for Bratu‐type and Lane‐Emden‐type problems

The present work deals with the construction of a high‐order efficient technique for solving one‐dimensional nonlinear Bratu‐type and Lane‐Emden‐type problems describing various physical, chemical and biological phenomena in applied science and engineering. Convergence analysis of the method is studied. Numerical experiments are carried out to demonstrate the efficiency, accuracy and performance of proposed method. The method is shown to have sixth order convergence. The computed results are compared with those obtained by other numerical techniques to justify the advantage of present scheme. It is shown that the proposed method is fast convergent and provides accurate solutions for Bratu‐type and Lane‐Emden‐type problems with low computational cost.

Optimal Algorithms for Nonlinear Equations with Applications and Their Dynamics

In the present work, we introduce two novel root‐finding algorithms for nonlinear scalar equations. Among these algorithms, the second one is optimal according to Kung‐Traub’s conjecture. It is established that the newly proposed algorithms bear the fourth‐ and sixth‐order of convergence. To show the effectiveness of the suggested methods, we provide several real‐life problems associated with engineering sciences. These problems have been solved through the suggested methods, and their numerical results proved the superiority of these methods over the other ones. Finally, we study the dynamics of the proposed methods using polynomiographs created with the help of a computer program using six cubic‐degree polynomials and then give a detailed graphical comparison with similar existing methods which shows the supremacy of the presented iteration schemes with respect to convergence speed and other dynamical aspects.

Efficient sixth order iterative method free from higher derivatives for nonlinear equations

In this paper, we proposed new iterative sixth order convergence method for solving nonlinear equations. The combination of the Taylor series and composition approach is used to derive the new method. Numerous methods have been developed by many researchers whenever the function’s second and higher order derivatives exist in the neighbourhood of the root. Computing the second and higher derivative of a function is a very cumbersome and time consuming task. In terms of low computation cost, the newly proposed method finds the best approximation to the root of non-linear equations by evaluating the function and its first derivative. The proposed method has been theoretically demonstrated to have sixth-order convergence. The proposed method has an efficiency index of 1.56. Several comparisons of the proposed method with the various existing iterative method of the same order have been performed on the number of problems. Finally, the computational results suggest that the newly proposed method is efficient compared to the well-known existing methods.

A New High-Order Jacobian-Free Iterative Method with Memory for Solving Nonlinear Systems

We used a Kurchatov-type accelerator to construct an iterative method with memory for solving nonlinear systems, with sixth-order convergence. It was developed from an initial scheme without memory, with order of convergence four. There exist few multidimensional schemes using more than one previous iterate in the very recent literature, mostly with low orders of convergence. The proposed scheme showed its efficiency and robustness in several numerical tests, where it was also compared with the existing procedures with high orders of convergence. These numerical tests included large nonlinear systems. In addition, we show that the proposed scheme has very stable qualitative behavior, by means of the analysis of an associated multidimensional, real rational function and also by means of a comparison of its basin of attraction with those of comparison methods.

On Various High-Order Newton-Like Power Flow Methods for Well and Ill-Conditioned Cases

Recently, the high-order Newton-like methods have gained popularity for solving power flow problems due to their simplicity, versatility and, in some cases, efficiency. In this context, recent research studied the applicability of the 4th order Jarrat’s method as applied to power flow calculation (PFC). Despite the 4th order of convergence of this technique, it is not competitive with the conventional solvers due to its very high computational cost. This paper addresses this issue by proposing two efficient modifications of the 4th order Jarrat’s method, which present the fourth and sixth order of convergence. In addition, continuous versions of the new proposals and the 4th order Jarrat’s method extend their applicability to ill-conditioned cases. Extensive results in multiple realistic power networks serve to sow the performance of the developed solvers. Results obtained in both well and ill-conditioned cases are promising.