Resumo em Inglês:

One of the ways that energy transports in fluid is electron thermal conduction. The aim of Inertial Confinement Fusion (ICF) is to show that the thermal conductivity is strongly dependent on temperature and the equation of heat condition is a nonlinear equation. In this article, we analyze the exact solutions of the nonlinear equation of heat conduction problem with variable transfer coefficients which is the problem of ICF.

Resumo em Inglês:

In this paper we suggest a fast numerical approach to treat problems of the hereditary linear viscoelasticity, which results in the system of elliptic partial differential equations in space variables, who's coefficients are Volterra integral operators of the second kind in time. We propose to approximate the relaxation kernels by the product of purely time- and space-dependent terms, which is achieved by their piecewise-polynomial space-interpolation. A priori error estimate was obtained and it was shown, that such approximation does not decrease the convergence order, when an interpolation polynomial is chosen of the same order as the shape functions for the spatial finite element approximation, while the computational effort is significantly reduced.

Resumo em Inglês:

A new iterative scheme that uses the residual vector as search direction is proposed and analyzed for solving large-scale nonsymmetric linear systems, whose matrix has a positive (or negative) definite symmetric part. It is closely related to Richardson's method, although the stepsize and some other new features are inspired by the success of recently proposed residual methods for nonlinear systems. Numerical experiments are included to show that, without preconditioning, the proposed scheme outperforms some recently proposed variations on Richardson's method, and competes with well-known and well-established Krylov subspace methods: GMRES and BiCGSTAB. Our computational experiments also show that, in the presence of suitable preconditioning strategies, residual iterative methods can be competitive, and sometimes advantageous, when compared with Krylov subspace methods.

Resumo em Inglês:

Restarting GMRES, a linear solver frequently used in numerical schemes, is known to suffer from stagnation. In this paper, a simple strategy is proposed to detect and avoid stagnation, without modifying the standard GMRES code. Numerical tests with the proposed modified GMRES(m) procedure for solving linear systems and also as part of an inexact Newton procedure, demonstrate the efficiency of this strategy.

Resumo em Inglês:

Let T be an arbitrary n × n matrix with real entries. We explicitly find the closest (in Frobenius norm) matrix A to T, where A is n × n with real entries, subject to the condition that A is ''generalized doubly stochastic'' (i.e. Ae = e and eT A = eT, where e = (1,1,...,1)T, although A is not necessarily nonnegative) and A has the same first moment as T (i.e. e1T Ae1 = e1T Te1). We also explicitly find the closest matrix A to T when A is generalized doubly stochastic has the same first moment as T and the same second moment as T (i.e. e1T A²e1 = e1T T²e1), when such a matrix A exists.

Resumo em Inglês:

In the present paper, we propose a numerical method for solving the coupled Lyapunov matrix equations A P + P A T + B B T = 0 and A T Q + Q A + C T C = 0 where A is an n ×n real matrix and B, C T are n × s real matrices with rank(B) = rank(C) = s and s << n . Such equations appear in control problems. The proposed method is a Krylov subspace method based on the nonsymmetric block Lanczos process. We use this process to produce low rank approximate solutions to the coupled Lyapunov matrix equations. We give some theoretical results such as an upper bound for the residual norms and perturbation results. By approximating the matrix transfer function F(z) = C (z In - A)-1 B of a Linear Time Invariant (LTI) system of order n by another one Fm(z) = Cm (z Im - Am)-1 Bm of order m, where m is much smaller than n , we will construct a reduced order model of the original LTI system. We conclude this work by reporting some numerical experiments to show the numerical behavior of the proposed method.