On the Stability Barrier of Hermite Type Discretizations of Advection Equations

ABSTRACT We establish a stability barrier for a class of high‐order Hermite‐type discretization of 1D advection equations underlying the hybrid‐variable (HV) and active flux (AF) methods. These methods approximate both cell averages and nodal solutions and evolve them in time simultaneously. The HV methods are known to be supraconvergent, providing that the discretization uses more unknowns in the upwind direction, similar to the “upwind condition” of classical finite‐difference schemes. Although it is well known that the stencil of finite‐difference methods could not be overly biased toward the upwind direction for stability reasons—a limitation known as the “stability barrier”—such a barrier has not been established for Hermite‐type methods. In this work, we first provide numerical evidence that a similar barrier exists for HV methods and conjecture a sharp bound on the stencil. We then prove the existence of a stability barrier by showing that the semi‐discretized HV methods become unstable when the stencil is sufficiently biased toward the upwind direction. Tighter barriers are subsequently proved using combinatorial tools, and finally, we extend the analysis to other Hermite‐type methods based on approximating nodal solutions and derivatives, such as those widely used in Hermite WENO methods.