Volume 24, Issue 1

In this paper, we derive a posteriori bounds of the difference between the exact solution of an elliptic boundary value problem with periodic coefficients and an abridged model, which follows from the homogenization theory. The difference is measured in terms of the energy norm of the basic problem and also in the combined primal–dual norm. Using the technique of functional type a posteriori error estimates, we obtain two-sided bounds of the modelling error, which depends only on known data and the solution of the homogenized problem. It is proved that the majorant with properly chosen arguments possesses the same convergence rate, which was established for the true error. Numerical tests confirm the efficiency of the estimates.

A family of real and analytical functions with values within the ring of M ( m , R ) is introduced. The solution for linear systems of differential equations is expressed as a series of Φ -functions. This new multistep method is defined for variable-step and variable-order, maintains the good properties of the Φ -function series method. It incorporates to compute the coefficients of the algorithm a recurrent algebraic procedure, based in the existing relation between the divided differences and the elemental and complete symmetrical functions. In addition, under certain hypotheses, the new multistep method calculates the exact solution of the perturbed problem. The new method is implemented in a computational algorithm which enables us to resolve in a general manner some physics and engineering IVP’s modeled by means systems of differential equations. The good behaviour and precision of the method is evidenced by contrasting the results with other-reputed algorithms and even with methods based on Scheifele’s G -functions.

Recently, Liu et al. [Q. B. Liu, G. L. Chen, J.Huang, On the parallel GSAOR method for block diagonallydominant matrices, Appl. Math. Comput . 215 (2009), 707–715] study the convergence of the parallel multisplitting generalized SAOR iterative methods based on the generalized AOR iterative method for solving a linear system whose coefficient matrix is a block diagonally dominant matrix or a generalized block diagonal dominant matrix. In this paper, we extend the domain of convergence from 1≤ω i < 2/(1+ μ 1 ( PMQ )) and 1≤ω i < 2/(1+ μ 2 ( PMQ )) to 0 < ω i < 2/(1 + μ 1 ( PMQ )) and 0 < ω i < 2/(1 + μ 2 ( PMQ )) for the parallel multisplitting generalized SAOR iterative methods.

A computational investigation of optimal control problems which are constrained by hyperbolic systems of conservation laws is presented. The general framework is to employ the adjoint-based optimization to minimize the cost functional of matching-type between the optimal and the target solution. Extension of the numerical schemes to second-order accuracy for systems for the forward and backward problem are applied. In addition a comparative study of two relaxation approaches as solvers for hyperbolic systems is undertaken. In particular optimal control of the 1-D Riemann problem of Euler equations of gas dynamics is studied. The initial values are used as control parameters. The numerical flow obtained by optimal initial conditions matches accurately with observations.