An analysis of a uniformly accurate difference method for a singular perturbation problem

Abstract

Abstract. It will be proven that an exponential tridiagonal difference scheme, when applied with a uniform mesh of size h to: euxx + b(x)ux « fix) for 0 < x < 1, b > 0, b and / smooth, e in (0, 1], and u(0) and u(\) given, is uniformly second-order accurate (i.e., the maximum of the errors at the grid points is bounded by Ch2 with the constant C independent of h and e). This scheme was derived by El-Mistikawy and Werle by a C1 patching of a pair of piecewise constant coefficient approximate differential equations across a common grid point. The behavior of the approximate solution in between the grid points will be analyzed, and some numerical results will also be given.

A Priori Estimates and Analysis of a Numerical Method for a Turning Point Problem

“…We next turn our attention to the Green's function for M in order to obtain the desired bounds on u2(x). 4 We now discuss some properties of the function g given by (4.4b). We write g(x)=/(0) + Kx), where Proof.…”

“…The discussion in the beginning of Section 2 of [4] shows that (3.1) has a unique solution in the sense just described. (The specific choices of P and Q given there are not required in the proof, which remains valid for the case of (3.1) if (3.2a) holds.)…”

“…(The specific choices of P and Q given there are not required in the proof, which remains valid for the case of (3.1) if (3.2a) holds.) From [6] or from Section 2 of [4] one has that at each interior grid point Xj a tridiagonal relationship of the form…”

“…is valid for the solution y of (3.1) where for each/ the r and 5 coefficients in (3.3) can be determined as follows [4]. Let P~ [P+] denote the value of P(x) on (xj_,, x¿)…”

“…Remark 2.2 of [4] shows that the linear system (3.1b), (3.3), (3.4) has a unique solution which may be calculated using simple tridiagonal Gaussian decomposition. Thus (3.1) yields a readily implementable algorithm for obtaining an approximation to the solution of (1.1).…”