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This paper presents the first systematic study on the fundamental problem ofseeking optimal cell average decomposition (OCAD), which arises fromconstructing efficient high-order bound-preserving (BP) numerical methodswithin Zhang--Shu framework. Since proposed in 2010, Zhang--Shu framework hasattracted extensive attention and been applied to developing many high-order BPdiscontinuous Galerkin and finite volume schemes for various hyperbolicequations. An essential ingredient in the framework is the decomposition of thecell averages of the numerical solution into a convex combination of thesolution values at certain quadrature points. The classic CAD originallyproposed by Zhang and Shu has been widely used in the past decade. However, thefeasible CADs are not unique, and different CAD would affect the theoretical BPCFL condition and thus the computational costs. Zhang and Shu only checked, forthe 1D $\mathbb P^2$ and $\mathbb P^3$ spaces, that their classic CAD based onthe Gauss--Lobatto quadrature is optimal in the sense of achieving the mildestBP CFL conditions.  In this paper, we establish the general theory for studying the OCAD problemon Cartesian meshes in 1D and 2D. We rigorously prove that the classic CAD isoptimal for general 1D $\mathbb P^k$ spaces and general 2D $\mathbb Q^k$ spacesof arbitrary $k$. For the widely used 2D $\mathbb P^k$ spaces, the classic CADis not optimal, and we establish the general approach to find out the genuineOCAD and propose a more practical quasi-optimal CAD, both of which provide muchmilder BP CFL conditions than the classic CAD. As a result, our OCAD andquasi-optimal CAD notably improve the efficiency of high-order BP schemes for alarge class of hyperbolic or convection-dominated equations, at little cost ofonly a slight and local modification to the implementation code.

On Optimal Cell Average Decomposition for High-Order Bound-Preserving Schemes of Hyperbolic Conservation Laws