Scielo RSS <![CDATA[Computational & Applied Mathematics]]> vol. 28 num. 1 lang. en <![CDATA[SciELO Logo]]> <![CDATA[<b>New approach for tanh and extended-tanh methods with applications on Hirota-Satsuma equations</b>]]> In this work, we establish new exact solutions for the Hirota-Satsuma equations. New approach for the tanh is used and extended tanh methods to construct traveling wave solutions in terms of a hyperbolic tangent functions. New families of solitary wave solutions and periodic solutions are also obtained for Hirota-Satsuma equations. Our approach is reduce the size of the computational adopted in other techniques without any conditions to apply on any system of partial differential equations. <![CDATA[<b>An inexact interior point proximal method for the variational inequality problem</b>]]> We propose an infeasible interior proximal method for solving variational inequality problems with maximal monotone operators and linear constraints. The interior proximal method proposed by Auslender, Teboulle and Ben-Tiba [3] is a proximal method using a distance-like barrier function and it has a global convergence property under mild assumptions. However, this method is applicable only to problems whose feasible region has nonempty interior. The algorithm we propose is applicable to problems whose feasible region may have empty interior. Moreover, a new kind of inexact scheme is used. We present a full convergence analysis for our algorithm. <![CDATA[<b>Numerical resolution of cone-constrained eigenvalue problems</b>]]> Given a convex cone K and matrices A and B, one wishes to find a scalar λ and a nonzero vector x satisfying the complementarity system K ∋ x ⊥(Ax-λ Bx) ∈ K+. This problem arises in mechanics and in other areas of applied mathematics. Two numerical techniques for solving such kind of cone-constrained eigenvalue problem are discussed, namely, the Power Iteration Method and the Scaling and Projection Algorithm. <![CDATA[<b>Unitary invariant and residual independent matrix distributions</b>]]> Define Z13 = A½Y(A½)H (A and Y are independent) and Z15 = B½Y(B½)H (B and Y are independent), where Y, A and B follow inverted complex Wishart, complex beta type I and complex beta type II distributions, respectively. In this article several properties including expected values of scalar and matrix valued functions of Z13 and Z15 are derived. <![CDATA[<b>Central schemes for porous media flows</b>]]> We are concerned with central differencing schemes for solving scalar hyperbolic conservation laws arising in the simulation of multiphase flows in heterogeneous porous media. We compare the Kurganov-Tadmor (KT) [3] semi-discrete central scheme with the Nessyahu-Tadmor (NT) [27] central scheme. The KT scheme uses more precise information about the local speeds of propagation together with integration over nonuniform control volumes, which contain the Riemann fans. These methods can accurately resolve sharp fronts in the fluid saturations without introducing spurious oscillations or excessive numerical diffusion. We first discuss the coupling of these methods with velocity fields approximated by mixed finite elements. Then, numerical simulations are presented for two-phase, two-dimensional flow problems in multi-scale heterogeneous petroleum reservoirs. We find the KT scheme to be considerably less diffusive, particularly in the presence of high permeability flow channels, which lead to strong restrictions on the time step selection; however, the KT scheme may produce incorrect boundary behavior. <![CDATA[<b>New versions of the Hestenes-Stiefel nonlinear conjugate gradient method based on the secant condition for optimization</b>]]> Based on the secant condition often satisfied by quasi-Newton methods, two new versions of the Hestenes-Stiefel (HS) nonlinear conjugate gradient method are proposed, which are descent methods even with inexact line searches. The search directions of the proposed methods have the form d k = - θkg k + βkHSd k-1, or d k = -g k + βkHSd k-1+ θky k-1. When exact line searches are used, the proposed methods reduce to the standard HS method. Convergence properties of the proposed methods are discussed. These results are also extended to some other conjugate gradient methods such as the Polak-Ribiére-Polyak (PRP) method. Numerical results are reported.