ORIGINAL PAPERS

Fully-Discrete Provably Lyapunov Consistent Discretizations for Convection-Diffusion-Reaction PDE Systems

  • Rasha Al Jahdali ,
  • David C. Del Rey Fernández ,
  • Lisandro Dalcin ,
  • Matteo Parsani
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  • 1. Computer Electrical and Mathematical Science and Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST), Thuwal, 23955-6900, Saudi Arabia;
    2. Department of Applied Mathematics, University of Waterloo, Waterloo, Canada;
    3. Physical Science and Engineering Division (PSE), King Abdullah University of Science and Technology (KAUST), Thuwal, 23955-6900, Saudi Arabia

Received date: 2024-10-28

  Revised date: 2025-04-27

  Accepted date: 2025-06-14

  Online published: 2026-09-17

Supported by

Open access publishing provided by King Abdullah University of Science and Technology (KAUST).

Abstract

Convection-diffusion-reaction equations are a class of second-order partial differential equations (PDEs) 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 simulations is crucial for accurate predictions, system analysis, and decision-making. This work focuses on the development of a comprehensive numerical framework for a class of convection-diffusion-reaction systems with a dissipative Lyapunov (or entropy or free energy) functional, Ṽ. This non-increasing Lyapunov functional is the driving quantity of the stability and properties of the system. We introduce a systematic methodology for constructing discretizations that mimic the stability analysis of the continuous model using Lyapunov’s direct method-type approach. The spatial algorithms are based on collocated discontinuous Galerkin (DG) methods with the summation-by-parts (SBP) property and the simultaneous approximation term (SAT) 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 PDEs 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.

Cite this article

Rasha Al Jahdali , David C. Del Rey Fernández , Lisandro Dalcin , Matteo Parsani . Fully-Discrete Provably Lyapunov Consistent Discretizations for Convection-Diffusion-Reaction PDE Systems[J]. Communications on Applied Mathematics and Computation, 2026 , 8(4) : 1604 -1652 . DOI: 10.1007/s42967-025-00514-1

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