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

Rasha Al Jahdali, David C. Del Rey Fernandez, Lisandro Dalcin, Matteo Parsani

Abstract

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.

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

Abstract

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.

Paper Structure

This paper contains 22 sections, 11 theorems, 151 equations, 18 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Lyapunov Consistency
  3. From ODEs to PDEs: Equilibrium points
  4. Spatial discretization: diffusion and convection
  5. Preliminaries
  6. Fully discrete schemes
  7. Relaxation Runge--Kutta methods
  8. Numerical Results
  9. Convergence Study
  10. Chemical reaction for the reversible dimerization model
  11. Preserving Lyapunov stability with $hp$-nonconforming discretization
  12. Discussion
  13. Global SBP operators for the first and second derivatives
  14. Computational space SBP operators
  15. Local Cartesian SBP operators
  16. ...and 7 more sections

Key Result

Theorem 1

Consider the existence of the equilibrium point $\bm{U}_{eq}$ of system eq:SEIR_con_diffW, (i.e., a point in phase space such that the right-hand side of system eq:SEIR_con_diffW is zero) where each of its components is bounded. Furthermore, assume that the following conditions are met: Then, defining the Lyapunov functional we have the following stability result:

Figures (18)

  • Figure 1: The reference element and its mapping to an element in a 3D mesh.
  • Figure 2: Temporal evolution of the maximum norm of the difference of the solution $(P,Q)$ and the equilibrium point $(P_{eq}, Q_{eq})$, Lyapunov functional, $\widetilde{V}$, and time derivative of the Lyapunov functional, $\frac{d \widetilde{V}}{d t}$, for the chemical reaction model reversible dimerization.
  • Figure 3: $t=0$
  • Figure 4: $t=0.127$
  • Figure 5: $t=6$
  • ...and 13 more figures

Theorems & Definitions (24)

  • Remark 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Definition 1
  • Theorem 3
  • proof
  • Definition 2
  • Theorem 4
  • ...and 14 more