Assess advantages of the alternative or hybrid hyperbolization for general linear scalar evolution PDEs

Determine whether adopting the hyperbolization formulation that replaces each lower-order term ∂_x^j u in the equation u_t + Σ_{j=0}^{m-1} α_j ∂_x^j u + σ_0 ∂_x^m u = 0 by the auxiliary variable q_j in the q_0-equation (the formulation denoted by equation (gen-hyp-B)), or employing a hybrid combination of this with the formulation that replaces ∂_x^j u by ∂_x^j q_{j-1} in the q_0-equation (equation (gen-hyp-A)), provides any advantage compared to using equation (gen-hyp-A).

Background

The paper introduces two natural hyperbolization choices for the general linear scalar evolution PDE u_t + Σ αj ∂_x^j u + σ_0 ∂_x^m u = 0. In formulation (gen-hyp-A), each lower-order derivative term ∂_x^j u in the q_0-equation is replaced by ∂_x^j q{j-1}. In formulation (gen-hyp-B), each such term is instead replaced by the auxiliary variable q_j.

The authors proceed with analysis and examples using (gen-hyp-A), but explicitly note uncertainty about whether (gen-hyp-B) or a hybrid of the two might be preferable. This raises a concrete comparative question about potential advantages (e.g., structural, stability, accuracy, or efficiency), which they explicitly mark as open.