A family of parametric schemes of arbitrary even order for solving nonlinear models

[EN] Many problems related to gas dynamics, heat transfer or chemical reactions are modeled by means of partial differential equations that usually are solved by using approximation techniques. When they are transformed in nonlinear systems of equations via a discretization process, this system is big-sized and high-order iterative methods are specially useful. In this paper, we construct a new family of parametric iterative methods with arbitrary even order, based on the extension of Ostrowski' and Chun's methods for solving nonlinear systems. Some elements of the proposed class are known methods meanwhile others are new schemes with good properties. Some numerical tests confirm the theoretical results and allow us to compare the numerical results obtained by applying new methods and known ones on academical examples. In addition, we apply one of our methods for approximating the solution of a heat conduction problem described by a parabolic partial differential equation.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P and FONDOCYT 2014-1C1-088 Republica Dominicana.Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vassileva, MP. (2017). A family of parametric schemes of arbitrary even order for solving nonlinear models. Journal of Mathematical Chemistry. 55(7):1443-1460. https://doi.org/10.1007/s10910-016-0723-7S14431460557R. Escobedo, L.L. Bonilla, Numerical methods for quantum drift-diffusion equation in semiconductor phisics. Math. Chem. 40(1), 3–13 (2006)S.J. Preece, J. Villingham, A.C. King, Chemical clock reactions: the effect of precursor consumtion. Math. Chem. 26, 47–73 (1999)H. Montazeri, F. Soleymani, S. Shateyi, S.S. Motsa, On a new method for computing the numerical solution of systems of nonlinear equations. J. Appl. Math. 2012 ID. 751975, 15 pages (2012)J.L. Hueso, E. Martínez, C. Teruel, Convergence, effiency and dinamimics of new fourth and sixth order families of iterative methods for nonlinear systems. J. Comput. Appl. Math. 275, 412–420 (2015)J.R. Sharma, H. Arora, Efficient Jarratt-like methods for solving systems of nonlinear equations. Calcolo 51, 193–210 (2014)X. Wang, T. Zhang, W. Qian, M. Teng, Seventh-order derivative-free iterative method for solving nonlinear systems. Numer. Algor. 70, 545–558 (2015)J.R. Sharma, H. Arora, On efficient weighted-Newton methods for solving systems of nonlinear equations. Appl. Math. Comput. 222, 497–506 (2013)A. Cordero, J.G. Maimó, J.R. Torregrosa, M.P. Vassileva, Solving nonlinear problems by Ostrowski-Chun type parametric families. J. Math. Chem. 53, 430–449 (2015)A.M. Ostrowski, Solution of equations and systems of equations (Prentice-Hall, Englewood Cliffs, New York, 1964)C. Chun, Construction of Newton-like iterative methods for solving nonlinear equations. Numer. Math. 104, 297–315 (2006)A. Cordero, J.L. Hueso, E. Martínez, J.R. Torregrosa, A modified Newton-Jarratt’s composition. Numer. Algor. 55, 87–99 (2010)J.M. Ortega, W.C. Rheinboldt, Iterative solution of nonlinear equations in several variables (Academic, New York, 1970)C. Hermite, Sur la formule dinterpolation de Lagrange. Reine Angew. Math. 84, 70–79 (1878)A. Cordero, J.R. Torregrosa, Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007

Arbitrary-Lagrangian-Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes

We present a new family of high order accurate fully discrete one-step Discontinuous Galerkin (DG) finite element schemes on moving unstructured meshes for the solution of nonlinear hyperbolic PDE in multiple space dimensions, which may also include parabolic terms in order to model dissipative transport processes. High order piecewise polynomials are adopted to represent the discrete solution at each time level and within each spatial control volume of the computational grid, while high order of accuracy in time is achieved by the ADER approach. In our algorithm the spatial mesh configuration can be defined in two different ways: either by an isoparametric approach that generates curved control volumes, or by a piecewise linear decomposition of each spatial control volume into simplex sub-elements. Our numerical method belongs to the category of direct Arbitrary-Lagrangian-Eulerian (ALE) schemes, where a space-time conservation formulation of the governing PDE system is considered and which already takes into account the new grid geometry directly during the computation of the numerical fluxes. Our new Lagrangian-type DG scheme adopts the novel a posteriori sub-cell finite volume limiter method, in which the validity of the candidate solution produced in each cell by an unlimited ADER-DG scheme is verified against a set of physical and numerical detection criteria. Those cells which do not satisfy all of the above criteria are flagged as troubled cells and are recomputed with a second order TVD finite volume scheme. The numerical convergence rates of the new ALE ADER-DG schemes are studied up to fourth order in space and time and several test problems are simulated. Finally, an application inspired by Inertial Confinement Fusion (ICF) type flows is considered by solving the Euler equations and the PDE of viscous and resistive magnetohydrodynamics (VRMHD).Comment: 39 pages, 21 figure

Spectral Methods for Numerical Relativity. The Initial Data Problem

Numerical relativity has traditionally been pursued via finite differencing. Here we explore pseudospectral collocation (PSC) as an alternative to finite differencing, focusing particularly on the solution of the Hamiltonian constraint (an elliptic partial differential equation) for a black hole spacetime with angular momentum and for a black hole spacetime superposed with gravitational radiation. In PSC, an approximate solution, generally expressed as a sum over a set of orthogonal basis functions (e.g., Chebyshev polynomials), is substituted into the exact system of equations and the residual minimized. For systems with analytic solutions the approximate solutions converge upon the exact solution exponentially as the number of basis functions is increased. Consequently, PSC has a high computational efficiency: for solutions of even modest accuracy we find that PSC is substantially more efficient, as measured by either execution time or memory required, than finite differencing; furthermore, these savings increase rapidly with increasing accuracy. The solution provided by PSC is an analytic function given everywhere; consequently, no interpolation operators need to be defined to determine the function values at intermediate points and no special arrangements need to be made to evaluate the solution or its derivatives on the boundaries. Since the practice of numerical relativity by finite differencing has been, and continues to be, hampered by both high computational resource demands and the difficulty of formulating acceptable finite difference alternatives to the analytic boundary conditions, PSC should be further pursued as an alternative way of formulating the computational problem of finding numerical solutions to the field equations of general relativity.Comment: 15 pages, 5 figures, revtex, submitted to PR

Comparative Study of Homotopy Analysis and Renormalization Group Methods on Rayleigh and Van der Pol Equations

A comparative study of the Homotopy Analysis method and an improved Renormalization Group method is presented in the context of the Rayleigh and the Van der Pol equations. Efficient approximate formulae as functions of the nonlinearity parameter $\varepsilon$ for the amplitudes $a(\varepsilon)$ of the limit cycles for both these oscillators are derived. The improvement in the Renormalization group analysis is achieved by invoking the idea of nonlinear time that should have significance in a nonlinear system. Good approximate plots of limit cycles of the concerned oscillators are also presented within this framework.Comment: 25 pages, 7 figures. Revised and upgraded: Differ Equ Dyn Syst, (26 July, 2015