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We study some theoretical aspects of $P_NP_M$ schemes, which are a novelclass of high order accurate reconstruction based discontinuous Galerkin (DG)schemes for hyperbolic conservation laws. The PNPM schemes store and evolve thediscrete solution $u_h$ under the form of piecewise polynomials of degree $N$,while piecewise polynomials $w_h$ of degree $M \geq N$ are used for thecomputation of the volume and boundary fluxes. The piecewise polynomials $w_h$are obtained from $u_h$ via a suitable reconstruction or recovery operator. The$P_NP_M$ approach contains high order finite volume methods ($N=0$) as well asclassical DG schemes ($N=M$) as special cases of a more general framework.Furthermore, for $N \neq M$ and $N>0$, it leads to a new intermediate class ofmethods, which can be denoted either as Hermite finite volume or asreconstructed DG methods. We show analytically why $P_NP_M$ methods for $N \neqM$ are, in general, not $L^2$-diminishing. To this end, we extend thewell-known cell entropy inequality and the following $L^2$ stability result ofJiang and Shu for DG methods (i.e. for $N=M$) to the general $P_NP_M$ case andidentify which part in the reconstruction step may cause the instability. Withthis insight we design a flux limiter that enforces a cell square entropyinequality and thus an $L^2$ stability condition for $P_NP_M$ schemes forscalar conservation laws in one space dimension. Furthermore, in this paper weprove existence and uniqueness of the solution of the $P_NP_M$ reconstructionoperator.

A square entropy stable flux limiter for $P_NP_M$ schemes