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Towards the ultimate conservative difference scheme. I: The quest of monotonicity. (English) Zbl 0255.76064
Proc. Third Int. Conf. Numer. Methods Fluid Mech., Univ. Paris 1972, 1, Lect. Notes Phys. 18, 163-168 (1973).
From the text: In the present paper I shall show that unconditional monotonicity involves the use of the ratio \(\Delta_{+\frac 12}a/\Delta_{-\frac 12}a\) in the dissipation coefficient. Unfortunately, a scheme based on Cl that includes this expression can never be conservative. To me, this represents the first valid reason to step up to a larger number of points in the basic cluster. Suitable five- and six-point clusters are shown in Fig. 2. Having no results ready for these more elaborate clusters, I shall occupy myself with the Lax-Wendroff scheme in the further sections of this paper. \(\ldots\)
The above results clearly demonstrate the usefulness of the smoothness monitor in constructing a monotonic scheme for a single nonlinear conservation law. The main point is now to design a monotonic scheme that is also conservative. The final step is applying it to a nonlinear system of conservation laws, in particular, the equations of ideal compressible flow.

76M20 Finite difference methods applied to problems in fluid mechanics
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