The solutions of two-point boundary-value problems often have boundary layers, narrow regions of sharp variation, that can occur in any part of the interval between the points. A finite difference method of numerical solution will generally require more closely spaced nodes in the boundary layers than elsewhere. An automatic method is needed for achieving the irregular spacing when the location of the boundary layer is not known in advance. Several automatic node-insertion or node-movement methods have been proposed. A new node-movement method is presented that is optimal under the criterion of producing the least sum of squares of the truncation errors at the nodes. For the Keller box scheme applied to a system of N coupled first-order differential equations this truncation-error minimizing (TEM) method increases the system size to N+6 equations. The campylotropic coordinate transformation method and other published methods based on heuristically derived monitor functions are node-movement methods that involve systems of only N+1 or N+2 first order equations. A comparison is made of the accuracies of several such methods and the TEM method in the solution of a standard problem.

A fully three-dimensional code has been written to compute the motion of a towed cable. The code is based on a robust and stable finite difference approximation to the differential equations derived from basic dynamics. A 3500-ft (1.07 km) cable pulled at 18.5 knots (34.3 km hr−1) through a circular turn of 700 yd (0.64 km) radius has been computed in about half of the real time of the maneuver. The computed displacements are close to the measured ones; the changes in depth are within 2%.

In cooperation with Catalysis Research Corporation (CRC) and the Institute of Gas Technology (IGT), SRI continued to provide some of the characterization data needed for the development of catalysts for the direct methanation of synthesis gas (syn gas) produced during coal gasification. Further understanding of catalyst properties was obtained by recognizing how the composition of the catalysts under study is related to their activity for methanation. This property manifests itself in terms of an acidity function that can be measured by 'titration' with a weak base, such as ammonia. The catalyst acidity so measured was found to parallel the catalyst's ability to dissociatively chemisorb hydrogen and to perform methanation of syn gas. In addition to high surface specific catalytic activity, thermal stability of the catalyst is desired so that the loss in effective surface area of the catalyst is small over most of the catalyst lifetime. The catalysts under study showed some tendancy to sinter at temperatures above 800K because of crystallite growth. Detailed measurements by X-ray photoelectron spectroscopy provided some insight into the sintering problem. Different methods were explored to reduce sintering.

In order to examine the relative contribution of surface and gas phase reactions to the exothermic conversion of a fuel-air gas mixture flowing over a catalytic surface, we have carried out a theoretical and experimental study of catalytic combustion under stagnation point flow conditions. In the presence of exothermic surface reaction the theoretical model provides an analytical solution; in the presence of both reaction modes, homogeneous and heterogeneous, computer solutions were obtained for the total heat flux at the surface and the distribution of temperature, reactant, and product concentrations in the stagnation point boundary layer. Having available kinetic data on the platinum-catalyzed oxidation of propane and on the gas phase reaction between propane and air, we selected this chemical fuel system for experimental and theoretical study. Experimentally the flow of the propane-air mixture (1 vol percent C3Hs) was directed at a quartz plate whose surface was coated with thin strips of vacuum-deposited platinum, that served both as catalyst and resistance thermometer. During an experiment the temperature of each strip was maintained constant (± 5 K.) by adjusting the electrical heat input as monitored by resistance measurements. By this procedure the heat released by exothermic reaction of the fuel-air mixture could be determined from the difference in electrical power required to keep each Pt strip at its original temperature. With the aid of the theoretical analysis we were able to compute the fractional contributions of catalytic and gas phase combustion to the total heat flux conducted to the catalytic surface. The relative contribution of each combustion mode' depended on catalyst activity, volumetric flow rate, and fuel-air ratio. Under our experimental conditions with transport limited surface reaction an increase in reactant flow rate enhances the surface-catalyzed contribution to the total heat release rate. The results obtained clearly indicate the conditions under which, either combustion mode predominates.

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTPhase boundaries for the carbon-hydrogen-oxygen system in equilibrium with carbides and oxides of iron and nickelS. Schechter and H. WiseCite this: J. Phys. Chem. 1979, 83, 16, 2107–2111Publication Date (Print):August 1, 1979Publication History Published online1 May 2002Published inissue 1 August 1979https://pubs.acs.org/doi/10.1021/j100479a011https://doi.org/10.1021/j100479a011research-articleACS PublicationsRequest reuse permissionsArticle Views82Altmetric-Citations4LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InRedditEmail Other access optionsSupporting Info (1)»Supporting Information Supporting Information Get e-Alerts

In boundary value problems for ordinary differential equations the finite-difference calculations for solutions having large variation over a narrow region often lose accuracy because of mesh irregularity, short steps being needed in the boundary layer and large steps elsewhere. This loss is eliminated by transformation to coordinates where a uniform mesh can be used. Several examples show that it is advantageous to take a linear combination of length and angular variation along the solution curve as the transformed coordinate for the one-dimensional case, only one-tenth as many nodes being needed in some cases as for other current methods.

Wall de-excitation of N2(v=1) was studied on a variety of different solid surfaces. Using 4880 Å laser radiation, we used the intensity of the Q branch of the anti-Stokes Raman scattering at 4382 Å to monitor the N2(v=1) concentration. The vibrationally excited nitrogen was produced by a thermal source and by a microwave discharge. The results were interpreted in terms of the two-dimensional diffusion equation with Poiseuille flow. The two sources of N2(v=1) gave somewhat different values for the wall deactivation coefficient γ. Furthermore, the results with the microwave source depended on the length of exposure of the surface to the afterglow. The observed differences are probably related to the fact that the microwave source also produces atoms and the thermal source does not. The lowest values of γ were recorded for quartz and Pyrex after 24 h of exposure to the afterglow. The results are interpreted in terms of a mechanism of heterogeneous vibrational de-excitation.

This chapter provides an overview on the choice of relaxation parameters for nonlinear problems. To obtain global convergence for relaxation methods, the relaxation parameters must generally be chosen from an interval dictated by iterate and smoothness of the functions involved. The chapter discusses an extension of these processes to those not requiring second derivative computation at each step. These methods are regarded as modified relaxation methods which include SOR-one-step secant, or Steffenson. Proofs of global convergence are indicated for these methods. A theorem is obtained for determining bounded level sets and the methods are illustrated by several examples taken from the variational calculus. For group relaxation, only the usual format of the iteration, where the elements of the Hessian are computed, are considered. The chapter explains several applications of the results available.

Previous article Next article On the Stability of Finite Difference MatricesK. W. Morton and S. SchechterK. W. Morton and S. Schechterhttps://doi.org/10.1137/0702010PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Mary Louise Buchanan, A necessary and sufficient condition for stability of difference schemes for initial value problems, J. Soc. Indust. Appl. Math., 11 (1963), 919–935 10.1137/0111067 MR0160324 (28:3537) 0221.65144 LinkISIGoogle Scholar[2] David H. Carlson and , Hans Schneider, Inertia theorems for matrices: the semidefinite case, J. Math. Anal. Appl., 6 (1963), 430–446 10.1016/0022-247X(63)90023-4 MR0148678 (26:6185) 0192.13402 CrossrefGoogle Scholar[3] Heinz-Otto Kreiss, Über Matrizen die beschränkte Halbgruppen erzeugen, Math. Scand., 7 (1959), 71–80 MR0110952 (22:1820) 0090.09801 CrossrefGoogle Scholar[4] Heinz-Otto Kreiss, Über die Stabilitätsdefinition für Differenzengleichungen die partielle Differentialgleichungen approximieren, Nordisk Tidskr. Informations-Behandling, 2 (1962), 153–181 MR0165712 (29:2992) 0109.34702 CrossrefGoogle Scholar[5] H. O. Kreiss, Über sachgemässe Cauehyprobleme für Systeme von linearen partiellen Differentialgleichungen, Kungl. Tekn. Högsk. Handl. Stockholm, 127 (1958), 1–30 0084.29801 Google Scholar[6] P. D. Lax and , R. D. Richtmyer, Survey of the stability of linear finite difference equations, Comm. Pure Appl. Math., 9 (1956), 267–293 MR0079204 (18,48c) 0072.08903 CrossrefISIGoogle Scholar[7] K. W. Morton, On a matrix theorem due to H. O. Kreiss, Comm. Pure Appl. Math., 17 (1964), 375–379 MR0170460 (30:698) 0146.13702 CrossrefISIGoogle Scholar[8] Olga Taussky, Matrices C with $C\sp{n}\rightarrow 0$, J. Algebra, 1 (1964), 5–10 10.1016/0021-8693(64)90003-1 MR0161865 (28:5069) 0126.02802 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails From Semidiscrete to Fully Discrete: Stability of Runge--Kutta Schemes by The Energy MethodDoron Levy and Eitan Tadmor4 August 2006 | SIAM Review, Vol. 40, No. 1AbstractPDF (856 KB)Stability Theory for Partial Difference OperatorsVidar Thomée18 July 2006 | SIAM Review, Vol. 11, No. 2AbstractPDF (3792 KB) Volume 2, Issue 1| 1965Journal of the Society for Industrial and Applied Mathematics Series B Numerical Analysis History Submitted:23 November 1964Published online:03 August 2006 InformationCopyright © 1965 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0702010Article page range:pp. 119-128ISSN (print):0887-459XISSN (online):2168-3581Publisher:Society for Industrial and Applied Mathematics

Analogous methods have been used in practice, with apparent success, on nonlinear problems as well. For the most part, these have not been justified mathematically and this work is an attempt to fill this gap. In particular it is shown that the relaxation methods yield solutions to problems arising from the minimization of certain convex functions. In practice, these functions are obtained by approximating multiple integrals in a calculus of variations problem. It is shown that an approximate Plateau problem may be solved by a successive displacements method, or a method analogous to Liebmann's method. We at the same time obtain an extension of a free steering theorem for positive definite symmetric matrices given as Theorem 4 of [3], and results of Ostrowski [2].

Article Free Access Share on Quasi-tridiagonal matrices and type-insensitive difference equations Author: Samuel Schechter New York University New York UniversityView Profile Authors Info & Claims ACM '59: Preprints of papers presented at the 14th national meeting of the Association for Computing MachinerySeptember 1959Pages 1–4https://doi.org/10.1145/612201.612239Published:01 September 1959Publication History 1citation135DownloadsMetricsTotal Citations1Total Downloads135Last 12 Months10Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

(1) H = i-î—¡-1 with 1 ^ i,j g n [a, — b,j let G = H~ = \dj\. (The indices i, j, k will range from 1 ton unless it is specified otherwise.) If, for some constant p, (2) o< bs » i1+ j 1+ p * 0, then H is a segment of the well known generalized Hubert matrix, and in this case formulas for c„ have been given by Savage and Lukacs [4], Smith [6] and Collar [1]. For (3) a, bj = i j + p, Linfoot and Shepherd [3] and Collar [2] give formulas for c<}and in both cases Collar exhibits diagonal matrices D, K such that G = DHTK. Collar [2] and Smith also evaluate the quantities 23 ca 1 23 ca ■ i,j j These authors make use, in most cases, of the formula for the determinant [5] II (ay ak)(h — bj) (4) det H = ^7-—Il (aj M i* or require the evaluation of certain involved series. The formulas of Collar and Smith are extended here to the general case (1). The method to be used does not depend on (4) but simply on Lagrange 's interpolation formula. Indeed (4) comes out as a by-product of formula (17) given below.

Communications on Pure and Applied MathematicsVolume 12, Issue 2 p. 313-335 Article Relaxation methods for linear equations† Samuel Schechter, Samuel SchechterSearch for more papers by this author Samuel Schechter, Samuel SchechterSearch for more papers by this author First published: May 1959 https://doi.org/10.1002/cpa.3160120208Citations: 16 † Presented, in part, at the Wayne State University Conference on Matrix Computations, September 3–6, 1957. The work for this paper was done at the AEC Computing and Applied Mathematics Center, Institute of Mathematical Sciences, under Contract AT(30-1)-1480 with the U. S. Atomic Energy Commission. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume12, Issue2May 1959Pages 313-335 RelatedInformation