Volume 1, Issue 4

In this paper we consider the one-dimensional nonlinear Schrödinger equation. The equation includes an absorption term, and the solution is periodically amplified in order to compensate the lose of the energy. The problem describes propa- gation of a signal in optical fibers. In our previous work we proved that the well-known Crank—Nicolson scheme is unconditionally unstable for this problem. We present in this paper two finite difference approximations. The first one is given by a modified Crank—Nicolson scheme and the second one is obtained by a splitting scheme. The stability and convergence of these schemes are proved. The results of numerical exper- iments are presented and discussed.

We propose a new discretization of an initial value problem for differen- tial equations of the first order in a Banach space with a strongly P-positive operator coefficient. Using the strong positiveness, we represent the solution as a Dunford- Cauchy integral along a parabola in the right half of the complex plane, then transform it into real integrals over (−∞,∞), and finally apply an exponentially convergent Sinc quadrature formula to this integral. The integrand values are the solutions of a finite set of elliptic problems with complex coefficients, which are independent and may be solved in parallel.

In this paper we study the convergence of a finite difference scheme that approximates the third boundary-value problem for an elliptic equation with variable coefficients on a unit square. An ”almost” second-order convergence rate estimate (with additional logarithmic multiplier) is obtained.

This paper studies an interior penalty discontinuous approximation of elliptic problems on nonmatching grids. Error analysis, interface domain decomposition type preconditioners, as well as numerical results illustrating both discretization errors and condition number estimates of the problem and reduced forms of it are presented.

In this paper we consider GMRES to solve finite-dimensional approxi- mations of a class of well-posed linear operator equations in Hilbert spaces. It is shown that the speed of convergence is superlinear. As a consequence we have that GMRES can be used as a fast solver of a fully discrete variant of the trigonometric Galerkin equations associated with periodic integral equations.

This paper is concerned with a high order difference scheme for a non- local boundary-value problem of parabolic equation. The integrals in the boundary equations are approximated by the composite Simpson rule. The unconditional solv- ability and L_∞ convergence of the difference scheme is proved by the energy method. The convergence rate of the difference scheme is second order in time and fourth order in space. Some numerical examples are provided to illustrate the convergence.