MATHEMATICSof computation VOLUME 46. NUMBER 173 JANUARY 11K6. PAGES 45-57 Convergence of Upwind Schemes for a Stationary Shock By Jens Lorenz* Abstract. A nonlinear first-order boundary value problem with discontinuous solutions is considered. It arises in the study of gasflow through a duct and allows, in general, for multiple solutions. New convergence results for three difference schemes are presented and the sharpness of numerical layers is established. For the EO-scheme, stability of a physically correct solution with respect to time evolution is shown. 1. Introduction. In this paper we analyze three difference schemes applied to a shock problem (1.1) -£f(u(x)) + b(x,u(x)) = 0, 0<x<l, «(0) = Y0,«(1) = Yi- Since the differential equation of first order is supplemented by two boundary conditions, we have to make precise what is meant by a solution of (1.1). In the case (1.2) bu(x,u)> p> 0 on[0,l]xR this is easily done: for all e > 0 the second-order problem (1.3) -eu" + /(«)' + b(x,u) = 0, 0<.x<l, h(0) ■= y0, u(l) = yx, is uniquely solvable and the solutions ue tend to a limit function U of bounded variation. U is considered as the solution of (1.1). Motivated by the considerations in [3], we are also interested in cases where the condition bu > 0 is violated. The one-dimensional duct flow equations for an inviscid gas can—for the stationary state—be reduced to a scalar equation for the velocity u, which has the form (1.1) (see, e.g., [16]). The condition bu > 0 is violated, e.g., for a converging-diverging duct. We make precise below what we understand by a solution of (1.1) in this case. Since (1.1) describes the stationary states of the hyperbolic problem u,+ f(u)x + b(x,u) = 0, 0 < x < 1, t > 0, (1.4) t/(0,0 = Yo. «(1,0-Yi. *>0, u(x,0) = (¡>(x), 0 «£* ^ 1, the question of stability with respect to time evolution is also of interest. If bu > 0, the solution U is stable; in the converging-diverging duct problem there are often two solutions, f/(1) and t/(2) with shocks; U{1) is unstable [3], It seems to be very likely that the solution i/<2), which has its shock in a region where bu > 0, is stable, but this has not yet been shown rigorously. Received June 7, 1984; revised January 31, 1985. 1980 Mathematics Subject Classification. Primary 65L10, 34B15. »Research supported by NSF Grant #DMS 83-12264. 1 14X6 American Mathematical Society 0025-5718/86 $1.00 + $.25 per page 45 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use