Fully discrete entropy and convergence theory

Develop a fully discrete entropy and convergence theory for the Operator DG schemes that incorporates the chosen time integrator, complementing the existing semi-discrete analysis.

Background

The paper establishes entropy inequalities, optimal error estimates, and strong convergence primarily for semi-discrete Operator DG schemes. Time discretization is used in numerical experiments, but its effects are not incorporated into the theoretical entropy and convergence analysis.

The unresolved problem is therefore to extend the analysis from the spatially semi-discrete setting to fully discrete schemes, including the conditions required of the time integrator to preserve entropy stability and convergence.

References

Several questions remain open. A fully discrete entropy and convergence theory that includes the time integrator would complement the present semi-discrete analysis.

— A class of high-order discontinuous-Galerkin methods satisfying infinitely many entropy conditions with provable error estimates and strong convergence for general nonlinear conservation laws  (2609.04687 - Wei et al., 4 Sep 2026) in Concluding remarks, Section 10

However, as noted in , whether Runge--Kutta methods without relaxation also preserve local energy/entropy balances remains an open question.

— An arbitrarily high-order accurate and locally energy/entropy-stable space-time method for a class of nonlinear wave systems  (2610.09797 - Dumbser et al., 7 Oct 2026) in Section 4, Fully discrete space–time method, immediately before the introduction of the CG–DG time discretization