A new iterative method is described for finding the real roots of nonlinear equations in R. Starting with a suitably chosen x 0, the method generates a sequence of iterates converging to the root. The convergence analysis is provided to establish its sixth order of convergence. The number of iterations and the total number of function evaluations used to get a simple root are taken as performan...

A second-order energy-preserving scheme is studied for the solution of the semilinear Cauchy Problem un uxx -uyy + u =0 (t > 0 ; x, y e R). Smooth data functions of compact support are prescribed at t = 0. On any time interval [0, 7"], second-order convergence (up to logarithmic corrections) to the exact solution is established in both the energy and uniform norms.

The goal of this article is to develop a new technique to obtain better asymptotic estimates for scalar conservation laws. General convex flux, f 00ðuÞX0; is considered with an assumption limu-0 uf 0ðuÞ=f ðuÞ 1⁄4 g41: We show that, under suitable conditions on the initial value, its solution converges to an N-wave in L norm with the optimal convergence order of Oð1=tÞ: The technique we use in t...

We examine the local convergence properties of pattern search methods, complementing the previously established global convergence properties for this class of algorithms. We show that the step-length control parameter which appears in the definition of pattern search algorithms provides a reliable asymptotic measure of first-order stationarity. This gives an analytical justification for a trad...

— We introduce low regularity exponential-type integrators for nonlinear Schrödinger equations for which first-order convergence only requires the boundedness of one additional derivative of the solution. More precisely, we will prove first-order convergence in H for solutions in H (r > d/2) of the derived schemes. This allows us lower regularity assumptions on the data than for instance requir...

We describe a method of random generation of the integrals A 1;2 (t; t + h) = Z t+h t Z s t dw 1 (r)dw 2 (s) ? Z t+h t Z s t dw 2 (r)dw 1 (s) together with the increments w 1 (t+h)?w 1 (t) and w 2 (t+h)?w 2 (t) of a two-dimensional Brownian path (w 1 (t);w 2 (t)). The method chosen is based on Marsaglia's `rectangle-wedge-tail' method, gen-eralised to higher dimensions. The motivation is the ne...

We develop an affine-scaling algorithm for box-constrained optimization which has the property that each iterate is a scaled cyclic Barzilai–Borwein (CBB) gradient iterate that lies in the interior of the feasible set. Global convergence is established for a nonmonotone line search, while there is local R-linear convergence at a nondegenerate local minimizer where the second-order sufficient op...

The eigenvalue problem of the Laplace-Beltrami operators on curved surfaces plays an essential role in the convergence analysis of the numerical simulations of some important geometric partial differential equations which involve this operator. In this note we shall combine the local tangential lifting (LTL) method with the configuration equation to develop a new effective and convergent algori...

A class Ψ of strictly concave and twice continuously differentiable functions ψ : R → R with particular properties is used for constraint transformation in the framework of a Nonlinear Rescaling (NR) method with “dynamic” scaling parameter updates. We show that the NR method is equivalent to the Interior Quadratic Prox method for the dual problem in a rescaled dual space. The equivalence is use...

We consider control-constrained linear-quadratic optimal control problems on evolving hypersurfaces in R. In order to formulate well-posed problems, we prove existence and uniqueness of weak solutions for the state equation, in the sense of vector-valued distributions. We then carry out and prove convergence of the variational discretization of a distributed optimal control problem. In the proc...