MATHEMATICS OF COMPUTATION, VOLUME 34, NUMBER 149 JANUARY 1980, PAGES 1-21 Monotone Difference Approximations for Scalar Conservation Laws By Michael G. Crandall and Andrew Majda* Abstract. A complete self-contained treatment of the stability and convergence proper- ties of conservation-form, monotone difference approximations to scalar conservation laws in several space variables is developed. In particular, the authors prove that gener- al monotone difference schemes always converge and that they converge to the physical weak solution satisfying the entropy condition. Rigorous convergence results follow for dimensional splitting algorithms when each step is approximated by a monotone difference scheme. The results are general enough to include, for instance, Godunov's scheme, the upwind scheme (differenced through stagnation points), and the Lax-Friedrichs scheme together with appropriate multi-dimensional generalizations. Introduction. Perhaps the simplest mathematical models exhibiting behavior typ- ical of that encountered in inviscid continuum mechanics are the initial-value problems for a scalar conservation law. These problems are of the form ( N \ (0 «f + L fiWxi = 0, fort>0,x = (xx,...,xN)GRN, (0.1) '=i ( (ii) u(x, 0) = u0(x), for x 6 R^, where the f¡ are smooth real-valued functions and u is a scalar. It is well known (see [14]) that even if the initial value «0 is smooth, the solution to (0.1) typically develops discontinuities as t increases to some t0 > 0 (i.e. shock waves form). Thus the differen- tial equation must be understood in a generalized or weak sense. However, there can be an infinite number of generalized solutions of (0.1) with the same initial data u0; and an additional principle, the entropy condition, is needed to select the unique "physical" weak solution (see [14]). The main new result of this work establishes the convergence of general conserva- tion-form, monotone difference approximations to (0.1) to the unique generalized solu- tion which satisfies the entropy condition. For notational simplicity in the sequel we restrict the presentation to the case N = 2 for the most part. The corresponding defi- nitions and results for the general case will be clear from this. For N = 2 we write (x, y) rather than (xx, x2). Selecting mesh sizes Ax, Ay, At > 0, the value Received February 12, 1979. AMS iMOS) subject classifications (1970). Primary 65M10, 65M05. Key words and phrases. Conservation laws, shock waves difference approximations, entropy conditions. •Partially supported by Sloan Foundation Fellowship. Sponsored by the United States Army under Contract No. DAAG29-75-C-0024. © 1980 American Mathematical Society 0025-5718/80/0000-0001/$06.25 1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use