SIAM Journal on Numerical Analysis

It is well known that the stability of the initial-boundary value problem for a scalar equation does not necessarily imply stability for a vector equation with a similar boundary treatment. In fact, we show that for any boundary treatment one can ...

The numerical solution of initial/boundary-value problems of the form \[ A(u,x,t)u_t + B(u,x,t)u_x = c(u,x,t)\] is considered. Particular emphasis is placed on the solution of problems with large gradients, e.g., shocks and boundary layers. A mesh-...

This paper discusses a class of compact second order accurate finite difference equations for mixed initial-boundary value problems for hyperbolic and convective-diffusion equations. Convergence is proved by means of energy arguments and both types of ...

In a 1965 paper [J. Math. and Phys., 44–45, pp. 301–311], Price, Varga and Warren deal with an observed oscillation in the numerical solution of \[ u_t = u_{xx} - ku_x ,\] subject to \[ u(x,0) = 0,\quad 0 < x < L,\]\[ u(0,t) = 1,\quad u_x (L,t) = 0,\...

We study the error due to the discretization in time of a parabolic problem by variable step size multistep methods. It is proved that the multistep methods remain stable when the rate of step changing is not too great. Error estimates are obtained for A-...

A new semidiscrete finite element method for the solution of second order nonlinear parabolic boundary value problems is formulated and analyzed. The test and trial spaces consist of discontinuous piecewise polynomial functions over quite general meshes ...

Galerkin and collocation approximations by trigonometric polynomials to the stationary Navier-Stokes equations with periodic boundary conditions are analyzed. Stability results and “optimal” rates of convergence for both velocity and pressure in Sobolev ...

Methods for the search of a maximum utilizing a sequence of majorants to generate a sequence of localizations are analyzed. Necessary and sufficient conditions for the convergence of this sequence are given. The case of Lipschitz-continuous functions is ...

Global convergence is established for Hirano’s modification of Newton’s method for solving algebraic equations. Specifically, it is proved that the amount of calculation involved in obtaining an improved approximation and the reduction rate of the ...

In [1], [2], [3], [4], a new method for solving systems of nonlinear equations was proposed and developed. The method associates a system of ordinary differential equations (odes) with the equations whose roots we are interested in and integrates the ...

First of all a method of collocation, which we call “classical” collocation, is described for the approximate solution of complete singular integral equations with Cauchy kernel taken over the arc $( - 1,1)$. Secondly we demonstrate that, under reasonable ...

In this paper we show that the optimal parameters for linear second-degree stationary iterative methods applied to nonsymmetric linear systems can be found by solving the same minimax problem used to find optimal parameters for the Chebyshev iteration. In ...

Some theorems (Pennacchi, Germain-Bonne, Smith and Ford) state that methods of a certain form which are exact on geometric sequences accelerate linear convergence. But no corresponding theorem is known for logarithmic convergence. Our study shows the ...

A successive interval test for existence and uniqueness of a solution to a nonlinear system and for convergence of iterative methods is given which is more powerful than the interval test introduced in [2] and [3].

The Moore test for nonlinear systems is improved in two directions. One presents the convergence test theorem under more general conditions. Another is a second-derivative test which reduces by half each edge of the test interval vector in the case of ...

This paper considers the problem of finding $\underline x $, $\overline x $ in $\mathbb{R}^n $ such that $\underline x \leqq G^{ - 1} h \leqq \overline x $ for any G in $\mathbb{R}^{n \times n} $ and h in $\mathbb{R}^n $ with $\underline A \leqq G \...