  This article aims at presenting a new local subcell monolithic
Discontinuous-Galerkin/Finite-Volume (DG/FV) convex property preserving scheme
solving system of conservation laws on 2D unstructured grids. This is known
that DG method needs some sort of nonlinear limiting to avoid spurious
oscillations or nonlinear instabilities which may lead to the crash of the
code. The main idea motivating the present work is to improve the robustness of
DG schemes, while preserving as much as possible its high accuracy and very
precise subcell resolution. To do so, a convex blending of high-order DG and
first-order FV scheme will be locally performed, at the subcell scale, where it
is needed. To this end, by means of the theory developed in our previous work,
we first recall that it is possible to rewrite DG scheme as a subcell FV scheme
on a subgrid provided with some specific numerical fluxes referred to as DG
reconstructed fluxes. Then, the monolithic DG/FV method will be defined as
following: each face of each subcell will be assigned with two fluxes, a
1st-order FV one and a high-order reconstructed one, that in the end will be
blended in a convex way. The goal is then to determine, through analysis,
optimal blending coefficients to achieve the desire properties. Numerical
results on various type problems will be presented to assess the very good
performance of the design method. A particular emphasis will be put on entropy
consideration. By means of this subcell monolithic framework, we will attempt
to address the following questions: is this possible through this monolithic
framework to ensure any entropy stability? what do we mean by entropy
stability? What is the cost of such constraints? Is this absolutely needed
while aiming for high-order accuracy?
