Derivative corrections for quadrature formulas

In this paper, we develop corrected quadrature formulas by approximating the derivatives of the integrand that appear in the asymptotic error expansion of the quadrature, using only the function values in the original quadrature rule. A higher order convergence is achieved without computing additional function values of the integrand.This author is in part supported by National Science Foundation under grant DMS-9504780 and by NASA-OAI Summer Faculty Fellowship (1995).     

Fourier analysis of a robust multigrid method for convection-diffusion equations

Summary. We consider a two-grid method for solving 2D convection-diffusion problems. The coarse grid correction is based on approximation of the Schur complement. As a preconditioner of the Schur complement we use the exact Schur complement of modified fine grid equations. We assume constant coefficients and periodic boundary conditions and apply Fourier analysis. We prove an upper bound for the spectral radius of the two-grid iteration matrix that is smaller than one and independent of the mesh size, the convection/diffusion ratio and the flow direction; i.e. we have a (strong) robustness result. Numerical results illustrating the robustness of the corresponding multigrid -cycle are given. Received October 14, 1994     

Reflecting Brownian motions: Quasimartingales and strong Caccioppoli sets

A notion ofstrong Caccioppoli set is defined for bounded Euclidean domains. It is shown that stationary (normally) reflecting Brownian motion on the closure of a bounded Euclidean domain is a quasimartingale on each compact time interval if and only if the domain is a strong Caccioppoli set. A similar result is shown to hold for symmetric reflecting diffusion processes.Research supported in part by NSF Grant DMS 91-01675.Research supported in part by NSF Grants DMS 86-57483 and 90-23335.     

Convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions

Summary. We prove convergence of a class of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions. The result is applied to the discontinuous Galerkin method due to Cockburn, Hou and Shu. Received April 15, 1993 / Revised version received March 13, 1995     

Evaluation of Chebyshev pseudospectral methods for third order differential equations

When the standard Chebyshev collocation method is used to solve a third order differential equation with one Neumann boundary condition and two Dirichlet boundary conditions, the resulting differentiation matrix has spurious positive eigenvalues and extreme eigenvalue already reaching O(N ⁵ for N = 64. Stable time-steps are therefore very small in this case. A matrix operator with better stability properties is obtained by using the modified Chebyshev collocation method, introduced by Kosloff and Tal Ezer [3]. By a correct choice of mapping and implementation of the Neumann boundary condition, the matrix operator has extreme eigenvalue less than O(N ⁴. The pseudospectral and modified pseudospectral methods are implemented for the solution of one-dimensional third-order partial differential equations and the accuracy of the solutions compared with those by finite difference techniques. The comparison verifies the stability analysis and the modified method allows larger time-steps. Moreover, to obtain the accuracy of the pseudospectral method the finite difference methods are substantially more expensive. Also, for the small N tested, N ⩽ 16, the modified pseudospectral method cannot compete with the standard approach. This revised version was published online in August 2006 with corrections to the Cover Date.     

Approximation of conformal mappings of annular regions

Summary. In this paper we examine the convergence rates in an adaptive version of an orthonormalization method for approximating the conformal mapping of an annular region onto a circular annulus. In particular, we consider the case where has an analytic extension in compl() and, for this case, we determine optimal ray sequences of approximants that give the best possible geometric rate of uniform convergence. We also estimate the rate of uniform convergence in the case where the annular region has piecewise analytic boundary without cusps. In both cases we also give the corresponding rates for the approximations to the conformal module of . Received February 2, 1996     

Inegalités Barlow-Yor pour les temps locaux d'intersection de deux mouvements Browniens plans

Summary We proveL _p> majorations for the suprema of the intersection local time of two planar Brownian motions. We deduce a sufficient condition on the correlation of two Brownian motions for their i.l.t. to be continuous.

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Inegalités Barlow-Yor pour les temps locaux d'intersection de deux mouvements Browniens plans

Résumé On prouve des majorations en normeL _p pour le suprema des temps locaux d'intersection de deux mouvements browniens plans, renormalisés ou non. On en déduit une condition suffisante sur la corrélation entre deux browniens pour que leur t.l.i. soient continus.

     

Asymptotic stability of viscous shock wave profiles for viscous conservation laws

This paper is concerned with the asymptotic stability of travelling wave solution to the two-dimensional steady isentropic irrotational flow with artificial viscosity. We prove that there exists a unique travelling wave solution up to a shift to the system if the end states satisfy both the Rankine–Hugoniot condition and Lax's shock condition, and that the travelling wave solution is stable if the initial disturbance is small.     

Krylov type subspace methods for matrix polynomials

We consider solving eigenvalue problems or model reduction problems for a quadratic matrix polynomial Iλ² − Aλ − B with large and sparse A and B. We propose new Arnoldi and Lanczos type processes which operate on the same space as A and B live and construct projections of A and B to produce a quadratic matrix polynomial with the coefficient matrices of much smaller size, which is used to approximate the original problem. We shall apply the new processes to solve eigenvalue problems and model reductions of a second order linear input-output system and discuss convergence properties. Our new processes are also extendable to cover a general matrix polynomial of any degree.     

Multiplicity of stationary solutions to the Euler-Poisson equations

Consider the system of Euler-Poisson as a model for the time evolution of gaseous stars through the self-induced gravitational force. We study the existence, uniqueness and multiplicity of stationary solutions for some velocity fields and entropy function that solve the conservation of mass and energy a priori. These results generalize the previous works on the irrotational or the rotational gaseous stars around an axis, and then they hold in more general physical settings. Under the assumption of radial symmetry, the monotonicity properties of the radius of the gas with respect to either the strength of the velocity field or the center density are also given which yield the uniqueness under some circumstances.