On the Convergence of Difference Approximations

We present a unified treatment of explicit in time, two-level, second-order resolution (SOR), total-variation diminishing (TVD), approximations to scalar conservation laws. The schemes are assumed only to have conservation form and incremental form. We introduce a modified flux and a viscosity coefficient and obtain results in terms of the latter. The existence of a cell entropy inequality is discussed and such an equality for all entropies is shown to imply that the scheme is an E scheme on monotone (actually more general) data, hence at most only first-order accurate in general. Convergence for TVD-SOR schemes approximating convex or concave conservation laws is shown by enforcing a single discrete entropy inequality. Introduction. Recently there has been an enormous amount of activity related to the construction and analysis of finite-difference approximations which approximate nonlinear hyperbolic systems of conservation laws and which are supposed to have the following properties: (1) Limit solutions which satisfy a geometric and/or analytic entropy condition. (2) A bound on the variation of the approximate solutions at least in the scalar and linear systems case. This bound is such as to imply the absence of spurious oscillations in the approximate solutions. (3) At least second-order accuracy in regions of smoothness, except for certain isolated points as described below. Some examples of the successful computational consequences of this activity can be found in the proceedings of the sixth AIAA Computational Fluid Dynamics Conference, and elsewhere; see, e.g., the bibliography in [21]. Some of the earliest work in the design of schemes having properties (2) and (3) above was done by Van Leer [27], [28]. There he introduced the concepts of flux limiters and higher-order Riemann solvers. Recently, Harten [10], [11] obtained conditions which he showed to be compatible with second-order accuracy, and which guarantee that a scalar one-dimensional scheme is TVD, that is, total-variation diminishing. He constructed a scheme having that property and formally extended it to systems using a field-by-field limiter and Roe's decomposition [22]. We would also like to mention the work of Boris and Book [1] concerning FCT schemes. They also used flux limiters to suppress oscillations in their schemes. Received May 21, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 65M10; Secondary 65M05. 'Research was supported in part by the National Aeronautics and Space Administration under NASA Contract No. NAS1-17070 while the authors were in residence at the Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, VA 23665. Additional support was provided by NASA Grant No. NAG 1-270, NSF Grant No. DMS-03294, ARO Grant No. DAAG-85-K-0190, and the U. S.-Israel BSF Grant No. 85-00346. ©1988 American Mathematical Society 0025-5718/88 $1.00 + $.25 per page