An efficient iterative method based on two-stage splitting methods to solve weakly nonlinear systems

[EN] In this paper, an iterative method for solving large, sparse systems of weakly nonlinear equations is presented. This method is based on Hermitian/skew-Hermitian splitting (HSS) scheme. Under suitable assumptions, we establish the convergence theorem for this method. In addition, it is shown that any faster and less time-consuming two-stage splitting method that satisfies the convergence theorem can be replaced instead of the HSS inner iterations. Numerical results, such as CPU time, show the robustness of our new method. This method is easy, fast and convenient with an accurate solution.The third and fourth authors have been partially supported by the Spanish Ministerio de Ciencia, Innovacion y Universidades PGC2018-095896-B-C22 and Generalitat Valenciana PROMETEO/2016/089.Amiri, A.; Darvishi, MT.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (2019). An efficient iterative method based on two-stage splitting methods to solve weakly nonlinear systems. Mathematics. 7(9):1-17. https://doi.org/10.3390/math7090815S1177

On the local convergence study for an efficient k-step iterative method

[EN] This paper is devoted to a family of Newton-like methods with frozen derivatives used to approximate a locally unique solution of an equation. The methods have high order of convergence but only using first order derivatives. Moreover only one LU decomposition is required in each iteration. In particular, the methods are real alternatives to the classical Newton method. We present a local convergence analysis based on hypotheses only on the first derivative. These types of local results were usually proved based on hypotheses on the derivative of order higher than two although only the first derivative appears in these types of methods (Bermficlez et al., 2012; Petkovic et al., 2013; Traub, 1964). We apply these methods to an equation related to the nonlinear complementarity problem. Finally, we find the most efficient method in the family for this problem and we perform a theoretical and a numerical study for it. (C) 2018 Elsevier B.V. All rights reserved.Research was supported in part by Programa de Apoyo a Ia investigacion de Ia fundacion Seneca-Agencia de Ciencia y Tecnologia de la Region de Murcia 19374/PI/14, by the project of Generalitat Valenciana Prometeo/2016/089 and the projects MTM2015-64382-P (MINECO/FEDER), MTM2014-52016-C2-1-P and MTM2014-52016-C2-2-P of the Spanish Ministry of Science and Innovation.Amat, S.; Argyros, IK.; Busquier Saez, S.; Hernández-Verón, MA.; Martínez Molada, E. (2018). On the local convergence study for an efficient k-step iterative method. Journal of Computational and Applied Mathematics. 343:753-761. https://doi.org/10.1016/j.cam.2018.02.028S75376134

Structure-Preserving Model Reduction of Physical Network Systems

This paper considers physical network systems where the energy storage is naturally associated to the nodes of the graph, while the edges of the graph correspond to static couplings. The first sections deal with the linear case, covering examples such as mass-damper and hydraulic systems, which have a structure that is similar to symmetric consensus dynamics. The last section is concerned with a specific class of nonlinear physical network systems; namely detailed-balanced chemical reaction networks governed by mass action kinetics. In both cases, linear and nonlinear, the structure of the dynamics is similar, and is based on a weighted Laplacian matrix, together with an energy function capturing the energy storage at the nodes. We discuss two methods for structure-preserving model reduction. The first one is clustering; aggregating the nodes of the underlying graph to obtain a reduced graph. The second approach is based on neglecting the energy storage at some of the nodes, and subsequently eliminating those nodes (called Kron reduction).</p

Astro-GRIPS, the General Relativistic Implicit Parallel Solver for Astrophysical Fluid Flows

In this work the development of the simulation code Astro-GRIPS, the General Relativistic Implicit Parallel Solver, is performed, which solves the three-dimensional axi-symmetric general relativistic hydrodynamic Euler or Navier-Stokes equations under the assumption of a fixed background metric of a Schwarzschild or Kerr black hole using time-implicit methods. It is an almost total re-write of an old spaghetti-code like serial Fortran 77 simulation program. By modernization and optimization it is now a modern, well structured, user-friendly, flexible and extensible simulation program written in Fortran 90/95. The finite volume discretization ensures conservation and the defect-correction iteration strategy is used to resolve the non-linearities of the equations. One can use a variety of solution procedures that range from purely explicit up to fully implicit schemes with up to third order spatial and second order temporal accuracy. The large sparse linear equation systems used for the implicit methods can be solved by the Black-White Line-Gauß-Seidel relaxation method (BW-LGS), the Approximate Factorization Method (AFM) or by Krylov Subspace Iterative methods like GMRES. The optimal solution method and the coupling of equations is problem-dependent. Optimizations in the matrix construction, the MPI-Parallelization for distributed memory machines and several Newtonian and relativistic tests were conducted successfully

Research in structural and solid mechanics, 1982

Advances in structural and solid mechanics, including solution procedures and the physical investigation of structural responses are discussed