Energy boundedness of the fully discrete LDG scheme

Establish rigorous energy boundedness for a fully discrete local discontinuous Galerkin scheme for the modified Camassa–Holm equation, extending the semi-discrete conservation and stability analysis to the fully discrete setting.

Background

The paper develops and analyzes a semi-discrete local discontinuous Galerkin method for the modified Camassa–Holm equation. It proves energy conservation for a central-flux scheme and energy dissipation for an upwind-flux scheme, but the time discretization is not included in the theoretical analysis.

A rigorous energy estimate for the fully discrete method would establish whether an appropriate time-stepping procedure preserves or bounds the discrete energy after temporal discretization. The authors note that relaxation Runge–Kutta methods may be used to achieve numerical conservation or dissipation, but a rigorous fully discrete energy-boundedness result is left unresolved. The concluding remarks reiterate that the fully discrete version is deferred to subsequent work.