In this paper we present numerical methods for the nonlinear Schrödinger equations (NLS) in the semiclassical regimes: iε uεt = − ε 2 ∆u + V (x)u + f(|u|)u, x ∈ R, with nonzero far-field conditions. A time-splitting cosine-spectral (TS-Cosine) method is presented when the nonzero far-field conditions are or can be reduced to homogeneous Neumann conditions, a time-splitting Chebyshev-spectral (T...

We introduce a three-step Chebyshev–Secant-type method (CSTM) with high efficiency index for solving nonlinear equations in a Banach space setting. We provide a semilocal convergence analysis for (CSTM)using recurrence relations. Numerical examples validating our theoretical results are also provided in this study. © 2011 Elsevier B.V. All rights reserved.

We present and analyze a new class of numerical methods for the solution of stiff stochastic differential equations (SDEs). These methods, called S-ROCK (for stochastic orthogonal Runge–Kutta Chebyshev), are explicit and of strong order 1 and possess large stability domains in the mean-square sense. For mean-square stable stiff SDEs, they are much more efficient than the standard explicit metho...

We describe a fast method for the evaluation of an arbitrary high-dimensional multivariate algebraic polynomial in Chebyshev form at the nodes of an arbitrary rank-1 Chebyshev lattice. Our main focus is on conditions on rank-1 Chebyshev lattices allowing for the exact reconstruction of such polynomials from samples along such lattices and we present an algorithm for constructing suitable rank-1...

Abstract We present numerical solutions for differential equations by expanding the unknown function in terms of Chebyshev polynomials and solving a system of linear equations directly for the values of the function at the extrema (or zeros) of the Chebyshev polynomial of order N (El-gendi’s method). The solutions are exact at these points, apart from round-off computer errors and the convergen...

This paper analyzes triangular finite elements for the preconditioning of Chebyshev collocation solutions of elliptic boundary value problems. Results are given for scalar model problems and for both Stokes and Navier-Stokes equations.