Fully-discrete provably Lyapunov consistent discretizations for convection-diffusion-reaction PDE systems

Convection-diffusion-reaction equations are a class of second-order partial differential equations widely used to model phenomena involving the change of concentration/population of one or more substances/species distributed in space. Understanding and preserving their stability properties in numerical simulation is crucial for accurate predictions, system analysis, and decision-making. This work presents a comprehensive framework for constructing fully discrete Lyapunov-consistent discretizations of any order for convection-diffusion-reaction models. We introduce a systematic methodology for constructing discretizations that mimic the stability analysis of the continuous model using Lyapunov's direct method. The spatial algorithms are based on collocated discontinuous Galerkin methods with the summation-by-parts property and the simultaneous approximation terms approach for imposing interface coupling and boundary conditions. Relaxation Runge-Kutta schemes are used to integrate in time and achieve fully discrete Lyapunov consistency. To verify the properties of the new schemes, we numerically solve a system of convection-diffusion-reaction partial differential equations governing the dynamic evolution of monomer and dimer concentrations during the dimerization process. Numerical results demonstrated the accuracy and consistency of the proposed discretizations. The new framework can enable further advancements in the analysis, control, and understanding of general convection-diffusion-reaction systems.