Multistep High-Order Methods for Nonlinear Equations Using Pade-Like Approximants

[EN] We present new high-order optimal iterativemethods for solving a nonlinear equation, f(x) = 0, by using Pade-like approximants. We compose optimal methods of order 4 with Newton's step and substitute the derivative by using an appropriate rational approximant, getting optimal methods of order 8. In the same way, increasing the degree of the approximant, we obtain optimal methods of order 16. We also perform different numerical tests that confirm the theoretical results.This work has been supported by Ministerio de Ciencia e Innovacion de Espana MTM2014-52016-C2-02-P and Generalitat Valenciana PROMETEO/2016/089.Cordero Barbero, A.; Hueso Pagoaga, JL.; Martínez Molada, E.; Torregrosa Sánchez, JR. (2017). Multistep High-Order Methods for Nonlinear Equations Using Pade-Like Approximants. Discrete Dynamics in Nature and Society. 1-6. https://doi.org/10.1155/2017/3204652S16Kung, H. T., & Traub, J. F. (1974). Optimal Order of One-Point and Multipoint Iteration. Journal of the ACM, 21(4), 643-651. doi:10.1145/321850.321860Petković, M. S., Neta, B., Petković, L. D., & Džunić, J. (2013). Basic concepts. Multipoint Methods, 1-26. doi:10.1016/b978-0-12-397013-8.00001-7Bi, W., Ren, H., & Wu, Q. (2009). Three-step iterative methods with eighth-order convergence for solving nonlinear equations. Journal of Computational and Applied Mathematics, 225(1), 105-112. doi:10.1016/j.cam.2008.07.004Cordero, A., Torregrosa, J. R., & Vassileva, M. P. (2011). Three-step iterative methods with optimal eighth-order convergence. Journal of Computational and Applied Mathematics, 235(10), 3189-3194. doi:10.1016/j.cam.2011.01.004Liu, L., & Wang, X. (2010). Eighth-order methods with high efficiency index for solving nonlinear equations. Applied Mathematics and Computation, 215(9), 3449-3454. doi:10.1016/j.amc.2009.10.040Sharma, J. R., & Sharma, R. (2009). A new family of modified Ostrowski’s methods with accelerated eighth order convergence. Numerical Algorithms, 54(4), 445-458. doi:10.1007/s11075-009-9345-5Cordero, A., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2010). New modifications of Potra–Pták’s method with optimal fourth and eighth orders of convergence. Journal of Computational and Applied Mathematics, 234(10), 2969-2976. doi:10.1016/j.cam.2010.04.009Wang, X., & Liu, L. (2010). New eighth-order iterative methods for solving nonlinear equations. Journal of Computational and Applied Mathematics, 234(5), 1611-1620. doi:10.1016/j.cam.2010.03.002Neta, B., & Petković, M. S. (2010). Construction of optimal order nonlinear solvers using inverse interpolation. Applied Mathematics and Computation, 217(6), 2448-2455. doi:10.1016/j.amc.2010.07.045Fidkowski, K. J., Oliver, T. A., Lu, J., & Darmofal, D. L. (2005). p-Multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier–Stokes equations. Journal of Computational Physics, 207(1), 92-113. doi:10.1016/j.jcp.2005.01.005Amat, S., & Busquier, S. (Eds.). (2016). Advances in Iterative Methods for Nonlinear Equations. SEMA SIMAI Springer Series. doi:10.1007/978-3-319-39228-8Bruns, D. D., & Bailey, J. E. (1977). Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state. Chemical Engineering Science, 32(3), 257-264. doi:10.1016/0009-2509(77)80203-0He, Y., & Ding, C. H. Q. (2001). The Journal of Supercomputing, 18(3), 259-277. doi:10.1023/a:1008153532043Revol, N., & Rouillier, F. (2005). Motivations for an Arbitrary Precision Interval Arithmetic and the MPFI Library. Reliable Computing, 11(4), 275-290. doi:10.1007/s11155-005-6891-yKing, R. F. (1973). A Family of Fourth Order Methods for Nonlinear Equations. SIAM Journal on Numerical Analysis, 10(5), 876-879. doi:10.1137/0710072Maheshwari, A. K. (2009). A fourth order iterative method for solving nonlinear equations. Applied Mathematics and Computation, 211(2), 383-391. doi:10.1016/j.amc.2009.01.047Weerakoon, S., & Fernando, T. G. I. (2000). A variant of Newton’s method with accelerated third-order convergence. Applied Mathematics Letters, 13(8), 87-93. doi:10.1016/s0893-9659(00)00100-2Cordero, A., & Torregrosa, J. R. (2007). Variants of Newton’s Method using fifth-order quadrature formulas. Applied Mathematics and Computation, 190(1), 686-698. doi:10.1016/j.amc.2007.01.06

Semilocal Convergence Analysis of an Iteration of Order Five Using Recurrence Relations in Banach Spaces

[EN] Semilocal convergence for an iteration of order five for solving nonlinear equations in Banach spaces is established under second-order Fr,chet derivative satisfying the Lipschitz condition. It is done by deriving a number of recurrence relations. A theorem for the existence-uniqueness along with the estimation of error bounds of the solution is established. Its R-order is shown to be equal to five. Both efficiency and computational efficiency indices are given. A variety of examples are worked out to show its applicability. In comparison to existing methods having similar R-orders, improved results in terms of computational efficiency index and error bounds are found using our methodology.The authors thank the referees for their valuable comments which have improved the presentation of the paper. The authors thankfully acknowledge the financial assistance provided by Council of Scientific and Industrial Research (CSIR), New Delhi, India.Singh, S.; Gupta, D.; Martínez Molada, E.; Hueso Pagoaga, JL. (2016). Semilocal Convergence Analysis of an Iteration of Order Five Using Recurrence Relations in Banach Spaces. Mediterranean Journal of Mathematics. 13(6):4219-4235. doi:10.1007/s00009-016-0741-5S42194235136Cordero A., Hueso J.L., Martinez E., Torregrosa J.R.: Increasing the convergence order of an iterative method for nonlinear systems. Appl. Math. Lett. 25, 2369–2374 (2012)Chen, L., Gu, C., Ma Y.: Semilocal convergence for a fifth order Newton’s method using Recurrence relations in Banach spaces. J. Appl. Math. 2011, 1–15 (2011)Wang X., Kou J., Gu C.: Semilocal convergence of a sixth order Jarrat method in Banach spaces. Numer. Algorithms 57, 441–456 (2011)Zheng L., Gu C.: Semilocal convergence of a sixth order method in Banach spaces. Numer. Algorithms 61, 413–427 (2012)Zheng L., Gu C.: Recurrence relations for semilocal convergence of a fifth order method in Banach spaces. Numer. Algorithms 59, 623–638 (2012)Proinov P.D., Ivanov S.I.: On the convergence of Halley’s method for multiple polynomial zeros. Mediterr. J. Math. 12, 555–572 (2015)Ezquerro, J.A., Hernández-Verón M.A.: On the domain of starting points of Newton’s method under center lipschitz conditions. Mediterr. J. Math. (2015). doi: 10.1007/s00009-015-0596-1Cordero A., Hernández-Verón M.A., Romero N., Torregrosa J.R.: Semilocal convergence by using recurrence relations for a fifth-order method in Banach spaces. J. Comput. Appl. Math. 273, 205–213 (2015)Parida P.K., Gupta D.K.: Recurrence relations for a Newton-like method in Banach spaces. J. Comput. Appl. Math. 206, 873–887 (2007)Hueso J.L., Martínez E.: Semilocal convergence of a family of iterative methods in Banach spaces. Numer. Algorithms 67, 365–384 (2014)Argyros, I.K., Hilout S.: Numerical methods in nonlinear analysis. World Scientific Publ. Comp., New Jersey (2013)Argyros, I.K., Hilout, S., Tabatabai, M.A.: Mathematical modelling with applications in biosciences and engineering. Nova Publishers, New York (2011)Argyros I.K., Khattri S.K.: Local convergence for a family of third order methods in Banach spaces. J. Math. 46, 53–62 (2004)Argyros I.K., Hilout A.S.: On the local convergence of fast two-step Newton-like methods for solving nonlinear equations. J. Comput. Appl. Math. 245, 1–9 (2013)Kantorovich, L.V., Akilov G.P.: Functional analysis. Pergamon Press, Oxford (1982)Argyros I.K., George S., Magreñán A.A.: Local convergence for multi-point-parametric Chebyshev-Halley-type methods of higher convergence order. J. Comput. Appl. Math. 282, 215–224 (2015)Argyros I.K., Magreñán A.A.: A study on the local convergence and the dynamics of Chebyshev-Halley-type methods free from second derivative. Numer. Algorithms 71, 1–23 (2015)Amat S., Hernández M.A., Romero N.: A modified Chebyshev’s iterative method with at least sixth order of convergence. Appl. Math. Comput. 206, 164–174 (2008)Chun, C., St $${\breve{a}}$$ a ˘ nic $${\breve{a}}$$ a ˘ , P., Neta, B.: Third-order family of methods in Banach spaces. Comput. Math. Appl. 61, 1665–1675 (2011)Ostrowski, A.M.: Solution of equations in Euclidean and Banach spaces, 3rd edn. Academic Press, New-York (1977)Jaiswal J.P.: Semilocal convergence of an eighth-order method in Banach spaces and its computational efficiency. Numer. Algorithms 71, 933–951 (2015)Traub, J.F.: Iterative methods for the solution of equations. Prentice-Hall, Englewood Cliffs (1964

Local convergence of a parameter based iteration with Holder continuous derivative in Banach spaces

[EN] The local convergence analysis of a parameter based iteration with Hölder continuous first derivative is studied for finding solutions of nonlinear equations in Banach spaces. It generalizes the local convergence analysis under Lipschitz continuous first derivative. The main contribution is to show the applicability to those problems for which Lipschitz condition fails without using higher order derivatives. An existence-uniqueness theorem along with the derivation of error bounds for the solution is established. Different numerical examples including nonlinear Hammerstein equation are solved. The radii of balls of convergence for them are obtained. Substantial improvements of these radii are found in comparison to some other existing methods under similar conditions for all examples considered.The authors thank the referees for their valuable comments which have improved the presentation of the paper. The authors thankfully acknowledge the financial assistance provided by Council of Scientific and Industrial Research (CSIR), New Delhi, India.Singh, S.; Gupta, DK.; Badoni, RP.; Martínez Molada, E.; Hueso Pagoaga, JL. (2017). Local convergence of a parameter based iteration with Holder continuous derivative in Banach spaces. CALCOLO. 54(2):527-539. doi:10.1007/s10092-016-0197-9S527539542Argyros, I.K., Hilout, S.: Numerical methods in nonlinear analysis. World Scientific Publ. Comp, New Jersey (2013)Argyros, I.K., Hilout, S., Tabatabai, M.A.: Mathematical modelling with applications in biosciences and engineering. Nova Publishers, New York (2011)Singh, S., Gupta, D.K., Martínez, E., Hueso, J.L.: Semilocal and local convergence of a fifth order iteration with Fréchet derivative satisfying Hölder condition. Appl. Math. Comput. 276, 266–277 (2016)Traub, J.F.: Iterative methods for the solution of equations. Prentice-Hall, Englewood Cliffs (1964)Rall, L.B.: Computational solution of nonlinear operator equations, reprint edn. R. E. Krieger, New York (2007)Cordero, A., Ezquerro, J.A., Hernández-Verón, M.A., Torregrosa, J.R.: On the local convergence of a fifth-order iterative method in Banach spaces. Appl. Math. Comput. 251, 396–403 (2015)Argyros, I.K., Hilout, A.S.: On the local convergence of fast two-step Newton-like methods for solving nonlinear equations. J. Comput. Appl. Math. 245, 1–9 (2013)Argyros, I.K., Behl, R., Motsa, S.S.: Local convergence of an efficient high convergence order method using hypothesis only on the first derivative. Algorithms 8, 1076–1087 (2015)Kantorovich, L.V., Akilov, G.P.: Functional analysis. Pergamon Press, Oxford (1982)Argyros, I.K., Magreñán, A.A.: A study on the local convergence and dynamics of Chebyshev-Halley-type methods free from second derivative. Numer. Algorithms 71, 1–23 (2016)Li, D., Liu, P., Kou, J.: An improvement ofthe Chebyshev-Halley methods free from second derivative. Appl. Math. Comput. 235, 221–225 (2014)Argyros, I.K., George, S.: Local convergence of deformed Halley method in Banach space under Holder continuity conditions. J. Nonlinear Sci. Appl. 8, 246–254 (2015)Argyros, I.K., Khattri, S.K.: Local convergence for a family of third order methods in Banach spaces. J. Math. 46, 53–62 (2014)Argyros, I.K., George, S., Magreñán, A.A.: Local convergence for multi-point-parametric Chebyshev-Halley-type methods of higher convergence order. J. Comput. Appl. Math. 282, 215–224 (2015)Argyros, I.K., George, S.: Local convergence of modified Halley-like methods with less computation of inversion. Novi. Sad. J. Math. 45, 47–58 (2015)Xiao, X.Y., Yin, H.W.: Increasing the order of convergence for iterative methods to solve nonlinear systems. Calcolo (2015). doi: 10.1007/s10092-015-0149-9Martínez, E., Singh, S., Hueso, J.L., Gupta, D.K.: Enlarging the convergence domain in local convergence studies for iterative methods in Banach spaces. Appl. Math. Comput. 281, 252–265 (2016

Implicit time integration for high-order compressible flow solvers

The application of high-order spectral/hp element discontinuous Galerkin (DG) methods to unsteady compressible flow simulations has gained increasing popularity. However, the time step is seriously restricted when high-order methods are applied to an explicit solver. To eliminate this restriction, an implicit high-order compressible flow solver is developed using DG methods for spatial discretization, diagonally implicit Runge-Kutta methods for temporal discretization, and the Jacobian-free Newton-Krylov method as its nonlinear solver. To accelerate convergence, a block relaxed Jacobi preconditioner is partially matrix-free implementation with a hybrid calculation of analytical and numerical Jacobian.The problem of too many user-defined parameters within the implicit solver is then studied. A systematic framework of adaptive strategies is designed to relax the difficulty of parameter choices. The adaptive time-stepping strategy is based on the observation that in a fixed mesh simulation, when the total error is dominated by the spatial error, further decreasing of temporal error through decreasing the time step cannot help increase accuracy but only slow down the solver. Based on a similar error analysis, an adaptive Newton tolerance is proposed based on the idea that the iterative error should be smaller than the temporal error to guarantee temporal accuracy. An adaptive strategy to update the preconditioner based on the Krylov solver’s convergence state is also discussed. Finally, an adaptive implicit solver is developed that eliminates the need for repeated tests of tunning parameters, whose accuracy and efficiency are verified in various steady/unsteady simulations. An improved shock-capturing strategy is also proposed when the implicit solver is applied to high-speed simulations. Through comparisons among the forms of three popular artificial viscosities, we identify the importance of the density term and add density-related terms on the original bulk-stress based artificial viscosity. To stabilize the simulations involving strong shear layers, we design an extra shearstress based artificial viscosity. The new shock-capturing strategy helps dissipate oscillations at shocks but has negligible dissipation in smooth regions.Open Acces

An Improved Ant Colony Algorithm for the Optimization of Skeletal Structures by the Proposed Sampling Search Space Method

Designing space is dramatically enlarged with optimization of structures based on ACO, regard to increasing section’s list. This problem decreases the speed of optimization in order to reach to optimum point and also increases local optimum probability, because determining suitable cross section process for each design variable in ACO depends on number of members in the list of section. Therefore, this paper by using partitioning the design space tries to decrease the probability of achieving local optimum during the process of structures optimum design by ACO and to increase the speed of convergence. In this regard, the list of section is divided to specific number of subsets inspired by meshing process in finite element. Then a member of each subset (in three case, maximum, middle and minimum of cross section) is defined as a representative of subset in a new list. Optimization process starts based on the new lit of section (global search). After specific number of repetitions, optimum design range for each variable will be determined. Afterward, variable section list is defined for each design variable related to result of previous step of process and based on subset of related variable. Finally, optimization process is continued based on the new list of section for each design variable to the end of process (local search). Proposal is studied in three cases and compared with common method in ACO and standard optimization examples in skeletal structures are used. Results show an increase in accuracy and speed of optimization according to cross section middle method (Case 2)

Numerical Analysis of Flux Reconstruction

High-order methods have become of increasing interest in recent years in computational physics. This is in part due to their perceived ability to, in some cases, reduce the computational overhead of complex problems through both an efficient use of computational resources and a reduction in the required degrees of freedom. One such high-order method in particular – Flux Reconstruction – is the focus of this thesis. This body of work relies and expands on the theoretical methods that are used to understand the behaviour of numerical methods – particularly related to their real-world application to industrial problems. The thesis begins by challenging some of the existing dogma surrounding computational fluid dynamics by evaluating the performance of high-order flux reconstruction. First, the use of the primitive variables as an intermediary step in the construction of flux terms is investigated. It is found that reducing the order of the flux function by using the conserved rather than primitive variables has a substantial impact on the resolution of the method. Critically, this is supported by a theoretical analysis, which shows that this mechanism of error generation becomes increasing important to consider as the order of accuracy increases. Next, the analysis of Flux Reconstruction was extended by analytically and numerically exploring the impact of higher dimensionality and grid deformation. It is found that both expanding and contracting grids – essential components of real-world domain decomposition – can cause dispersion overshoot in two dimensions, but that FR appears to suffer less that comparable Finite Difference approaches. Fully discrete analysis is then used to show that, depending on the correction function, small perturbations in incidence angle can cause large changes in group velocity. The same analysis is also used to theoretically demonstrate that Discontinuous Galerkin suffers less from dispersion error than Huynh’s FR scheme – a phenomenon that has previously been observed experimentally, but not explained theoretically. This thesis concludes with the presentation of a robust theoretical underpinning for determining stable correction functions for FR. Three new families of correction functions are presented, and their properties extensively explored. An important theoretical finding is introduced – that stable correction functions are not defined uniquely be a norm. As a result, a generalised approach is presented, which is able to recover all previously defined correction functions, but in some instances via a different norm to their original derivation. This new super-family of correction functions shows considerable promise in increasing temporal stability limits, reducing dispersion when fully discretised, and increasing the rate of convergence. Taken altogether, this thesis represents a considerable advance in the theoretical characterisation and understanding of a numerical method – one which, it has been shown, has enormous potential for forming the heart of future computational physics codes

Solving nonlinear problems by Ostrowski Chun type parametric families

In this paper, by using a generalization of Ostrowski' and Chun's methods two bi-parametric families of predictor-corrector iterative schemes, with order of convergence four for solving system of nonlinear equations, are presented. The predictor of the first family is Newton's method, and the predictor of the second one is Steffensen's scheme. One of them is extended to the multidimensional case. Some numerical tests are performed to compare proposed methods with existing ones and to confirm the theoretical results. We check the obtained results by solving the molecular interaction problem.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and FONDOCYT, Republica Dominicana.Cordero Barbero, A.; Maimo, J.; Torregrosa Sánchez, JR.; Vassileva, M. (2015). Solving nonlinear problems by Ostrowski Chun type parametric families. Journal of Mathematical Chemistry. 53(1):430-449. https://doi.org/10.1007/s10910-014-0432-zS430449531M.S. Petkovic̀, B. Neta, L.D. 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