L ∞ stability of the MUSCL methods

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We present a general L ∞ stability result for generic finite volume methods coupled with a large class of reconstruction for hyperbolic scalar equations. We show that the stability is obtained if the reconstruction respects two fundamental properties: the convexity property and the sign inversion property. We also introduce a new MUSCL technique named the multislope MUSCL technique based on the approximations of the directional derivatives in contrast to the classical piecewise reconstruction, the so-called monoslope MUSCL technique, based on the gradient reconstruction. We show that under specific constraints we shall detail, the two MUSCL reconstructions satisfy the convexity and sign inversion properties and we prove the L ∞ stability.