MATHEMATICS OF COMPUTATION VOLUME 47, NUMBER 176 OCTOBER 1986, PAGES 511-535 The Numerical Solution of Second-Order Boundary Value Problems on Nonuniform Meshes By Thomas A. Manteuffel and Andrew B. White, Jr. Abstract. In this paper, we examine the solution of second-order, scalar boundary value problems on nonuniform meshes. We show that certain commonly used difference schemes yield second-order accurate solutions despite the fact that their truncation error is of lower order. This result illuminates a limitation of the standard stability, consistency proof of convergence for difference schemes defined on nonuniform meshes. A technique of reducing centered-difference approximations of first-order systems to discretizations of the underlying scalar equation is developed. We treat both vertex-centered and cell-centered difference schemes and indicate how these results apply to partial differential equations on Cartesian product grids. 1. Introduction. Much attention has been paid to the numerical solution of second-order differential equations on nonuniform meshes. To begin, we consider the solution of the linear, two-point boundary value problem (1.1) y" + a(x)y' + b(x)y=f(x), xe=(0,l), (1.2a) b^O) + boxy'(0) = b01, (1.2b) bxoy(l) + bxxy'(l) = bX2, through three-point (compact-as-possible in the sense of Kreiss [15]) difference schemes on a mesh {x^^Zq. The functions a, b, and / are assumed to be smooth. The standard mesh spacing will be written as A; = x¡ — x¡_v and we will denote a function evaluated at the mesh points by subscripts, y¡, and the vector with these entries as Y. The ith component of Y may also be written (Y)¡. We will make liberal use of the generic constant C and of A = max {A,}. An example of such a difference scheme is that used by Pearson [21] in his classic work on singular perturbations and by Denny and Landis [5] and de Rivas [6] in their work on mesh-selection techniques. In this method, each term in (1.1) is approximated separately on the stencil (x - A,,x,x + A,+1). Approximations to the derivatives are (1.3a) A,(A, + A,+1)' A,A,+1' (A, + A,+1)A,+1 = y" + ±(A, +x-A,)y'" +0(A2) y,-i y, yi+i Received January 26, 1984; revised May 17, 1985. 1980 Mathematics Subject Classification. Primary 65L05, 65L10;Secondary 65L25, 40A30. Key words and phrases. Boundary value problems, compact difference schemes, irregular grids, nonuniform mesh, difference quotients, truncation error. ©1986 American Mathematical Society 0025-5718/86 $1.00 + $.25 per page 511 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use