L∞ stability of the MUSCL methods

We present a general L stability result for a generic finite volume method for hyperbolic scalar equations coupled with a large class of reconstruction. 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, the multislope MUSCL technique, based on the approximations of the directional derivative in contrast to the classical piecewise reconstruction, the 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.