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 (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,

\tilde{V}

. 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.

Keywords

Convection-diffusion-reaction equations / Lyapunov functionals / Summation-by-parts (SBP) operators / Relaxation Runge-Kutta schemes / Fully discrete Lyapunov-consistent discretizations / 93D05 / 65L06 / 35K57 / 65N12 / 65N35 / 92D30

References

[1] Al Jahdali, R., Dalcin, L., Parsani, M.: On the performance of relaxation and adaptive explicit Runge-Kutta schemes for high-order compressible flow simulations. J. Comput. Phys. 464, 111333 (2022) https://doi.org/10.1016/j.jcp.2022.111333

[2] Al Jahdali R, Rey Fernández DC, Dalcin L, Sayyari M, Markowich P, Parsani M. Fully discrete Lyapunov consistent discretizations for parabolic reaction-diffusion equations with r species. Commun. Appl. Math. Comput., 2024.

[3] Alhumaizi K, Henda R, Soliman M. Numerical analysis of a reaction-diffusion-convection system. Comput. Chem. Eng., 2003, 27(4): 579-594.

[4] Ålund, O., Nordström, J.: Encapsulated high order difference operators on curvilinear non-conforming grids. J. Comput. Phys. 385, 209–224 (2019) https://doi.org/10.1016/j.jcp.2019.02.007

[5] Benzi R, Nelson DR, Shankar S, Toschi F, Zhu X. Spatial population genetics with fluid flow. Rep. Prog. Phys., 2022, 85(9. 096601

[6] Bertsekas DP. Nonlinear programming. J. Oper. Res. Soc., 1997, 48(3): 334-334.

[7] Besançon, G., Pham, T.V., Georges, D.: Robust state estimation for a class of convection-diffusion-reaction systems. IFAC Proceedings Volumes 46(26), 203–208 (2013). https://doi.org/10.3182/20130925-3-FR-4043.00012

[8] Bressan, A.: Hyperbolic Systems of Conservation Laws: the One-Dimensional Cauchy Problem vol. 20. Oxford University Press, USA (2000). https://doi.org/10.1093/oso/9780198507000.001.0001

[9] Brezzi, F., Ushiki, S., Fujii, H.: “Real” and “ghost” bifurcation dynamics in difference schemes for ODEs. In: Küpper, T., Mittelmann, H.D., Weber, H. (eds) Numerical Methods for Bifurcation Problems. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol. 70, pp. 79–104. Birkhäuser, Basel (1984). https://doi.org/10.1007/978-3-0348-6256-1_6

[10] Buonomo, B., Rionero, S.: On the Lyapunov stability for SIRS epidemic models with general nonlinear incidence rate. Appl. Math. Comput. 217(8), 4010–4016 (2010). https://doi.org/10.1016/j.amc.2010.10.007

[11] Burton, T.A.: Stability by Fixed Point Theory for Functional Differential Equations, vol. 1. Courier Corporation (2013)

[12] Butcher JC. Numerical Methods for Ordinary Differential Equations. John Wiley & Sons, Ltd, 2008.

[13] Carpenter MH, Fisher TC, Nielsen EJ, Frankel SH. Entropy stable spectral collocation schemes for the Navier-Stokes equations: discontinuous interfaces. SIAM J. Sci. Comput., 2014, 36(5): 835-867.

[14] Carpenter, M.H., Gottlieb, D., Abarbanel, S.: Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes. J. Comput. Phys. 111(2), 220–236 (1994). https://doi.org/10.1006/jcph.1994.1057

[15] Carpenter MH, Nordström J, Gottlieb D. A stable and conservative interface treatment of arbitrary spatial accuracy. J. Comput. Phys., 1999, 148(2): 341-365.

[16] Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. II. Interscience Publishers, New York, NY (1962). https://doi.org/10.1002/9783527617234

[17] Crean, J., Hicken, J.E., Del Rey Fernández, D.C., Zingg, D.W., Carpenter, M.H.: Entropy-stable summation-by-parts discretization of the Euler equations on general curved elements. J. Comput. Phys. 356, 410–438 (2018) https://doi.org/10.1016/j.jcp.2017.12.015

[18] Curtain, R., Zwart, H.: Introduction to Infinite-Dimensional Systems Theory: a State-Space Approach. Springer, New York (2020). https://doi.org/10.1007/978-1-0716-0590-5

[19] Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics. Springer, New York (2016). https://doi.org/10.1007/978-3-662-49451-6

[20] Del Rey Fernández, D.C., Boom, P.D., Carpenter, M.H., Zingg, D.W.: Extension of tensor-product generalized and dense-norm summation-by-parts operators to curvilinear coordinates. J. Sci. Comput. 80, 1957–1996 (2019) https://doi.org/10.1007/s10915-019-01011-3

[21] Del Rey Fernández, D.C., Carpenter, M.H., Dalcin, L., Zampini, S., Parsani, M.: Entropy stable h/p-nonconforming discretization with the summation-by-parts property for the compressible Euler and Navier-Stokes equations. SN Partial Differ. Equ. Appl. 1(2), 1–54 (2020). https://doi.org/10.1007/s42985-020-00009-z

[22] Del Rey Fernández DC, Hicken JE, Zingg DW. Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations. Comput. Fluids, 2014, 95(22): 171-196.

[23] Del Rey Fernández, D.C., Hicken, J.E., Zingg, D.W.: Simultaneous approximation terms for multidimensional summation-by-parts operators. J. Sci. Comput. 75, 83–110 (2018) https://doi.org/10.1007/s10915-017-0523-7

[24] Drábek, P., Zahradníková, M.: Traveling waves in reaction-diffusion-convection equations with combustion nonlinearity. Nonlinear Anal.: Real World Appl. 84, 104283 (2025). https://doi.org/10.1016/j.nonrwa.2024.104283

[25] Duru, K., Gabriel, A.-A., Kreiss, G.: On energy stable discontinuous Galerkin spectral element approximations of the perfectly matched layer for the wave equation. Comput. Method. Appl. Mech. Eng. 350, 898–937 (2019) https://doi.org/10.1016/j.cma.2019.02.036

[26] Feireisl E, Laurençot P, Petzeltová H. On convergence to equilibria for the Keller-Segel chemotaxis model. J. Differ. Equ., 2007, 236(2): 551-569.

[27] Fernandez, D.D.R., Carpenter, M.H., Dalcin, L., Fredrich, L., Rojas, D., Winters, A., Gassner, G., Zampini, S., Parsani, M.: Entropy stable nonconforming discretizations with the summation-by-parts property for curvilinear coordinates. NASA/TM-2020-220574 (2020)

[28] Fisher TC, Carpenter MH. High-order entropy stable finite difference schemes for nonlinear conservation laws: finite domains. J. Comput. Phys., 2013, 252(1): 518-557.

[29] Fisher TC, Carpenter MH, Nordström J, Yamaleev NK. Discretely conservative finite-difference formulations for nonlinear conservation laws in split form: theory and boundary conditions. J. Comput. Phys., 2013, 234(1): 353-375.

[30] Friedrich, L., Winters, A.R., Del Rey Fernández, D.C., Gassner, G.J., Parsani, M., Carpenter, M.H.: An entropy stable h/p non-conforming discontinuous Galerkin method with the summation-by-parts property. J. Sci. Comput. 77, 689–725 (2018) https://doi.org/10.1007/s10915-018-0733-7

[31] Garzón-Alvarado DA, Galeano CH, Mantilla JM. Computational examples of reaction-convection-diffusion equations solution under the influence of fluid flow: first example. Appl. Math. Model., 2012, 36(10): 5029-5045.

[32] Greenberg JM, Ta Tsien L. The effect of boundary damping for the quasilinear wave equation. J. Differ. Equ., 1984, 52(1): 66-75.

[33] Griffiths D, Sweby P, Yee HC. On spurious asymptotic numerical solutions of explicit Runge-Kutta methods. IMA J. Numer. Anal., 1992, 12(3): 319-338.

[34] Gustafsson, B., Kreiss, H.O., Oliger, J.: Time-Dependent Problems and Difference Methods. John Wiley & Sons, Inc. (2013). https://doi.org/10.1002/9781118548448

[35] Hairer E, Iserles A, Sanz-Serna JM. Equilibria of Runge-Kutta methods. Numer. Math., 1990, 58(1): 243-254.

[36] Harten A. On the symmetric form of systems of conservation laws with entropy. J. Comput. Phys., 1983, 49(1): 151-164.

[37] Hicken, J.E., Del Rey Fernández, D.C., Zingg, D.W.: Multidimensional summation-by-part operators: general theory and application to simplex elements. SIAM J. Sci. Comput. 38(4) (2016) https://doi.org/10.1137/15M1038360

[38] Iserles, A.: Stability and dynamics of numerical methods for nonlinear ordinary differential equations. IMA J. Numer. Anal. 10, 1–30 (1990) https://doi.org/10.1093/imanum/10.1.1

[39] Ketcheson DI. Relaxation Runge-Kutta methods: conservation and stability for inner-product norms. SIAM J. Numer. Anal., 2019, 57(6): 2850-2870.

[40] Khalil HKNonlinear Systems, 20023Upper Saddle River, NJPrentice-Hall

[41] Knoll DA. An improved convection scheme applied to recombining divertor plasma flows. J. Comput. Phys., 1998, 142(2): 473-488.

[42] Komornik, V.: Exact Controllability and Stabilization: the Multiplier Method, vol. 39. Wiley, Chichester (1994)

[43] Korobeinikov A, Wake GC. Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models. Appl. Math. Lett., 2002, 15(8): 955-960.

[44] Kreiss, H.-O., Scherer, G.: Finite element and finite difference methods for hyperbolic partial differential equations. In: Boor, C. (ed.) Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 195–212. Academic Press (1974). https://doi.org/10.1016/B978-0-12-208350-1.50012-1

[45] Krstic M, Guo B-Z, Balogh A, Smyshlyaev A. Output-feedback stabilization of an unstable wave equation. Automatica, 2008, 44(1): 63-74.

[46] Krstic M, Smyshlyaev A. Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays. Syst. Control Lett., 2008, 57(9): 750-758.

[47] LaSalle, J.P., Artstein, Z.: The Stability of Dynamical Systems. CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (1976)

[48] LaSalle, J.P., Lefschetz, S.: Stability by Liapunov’s Direct method: With Applications. Mathematics in Science and Engineering: a Series of Monographs and Textbooks. Academic Press, New York (1961)

[49] Lax PD. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. SIAM, 1973.

[50] Luenberger, D.G., Ye, Y.: Linear and Nonlinear Programming, vol. 2. Springer, New York (1984)

[51] Luo Z-H, Guo B-Z, Morgül Ö. Stability and Stabilization of Infinite Dimensional Systems with Applications. Springer, 1999.

[52] Mattsson K. Boundary procedures for summation-by-parts operators. J. Sci. Comput., 2003, 18(1): 133-153.

[53] Mattsson K, Ham F, Iaccarino G. Stable boundary treatment for the wave equation on second-order form. J. Sci. Comput., 2009, 41(3): 366-383.

[54] Merriam, M.L.: An Entropy-Based Approach to Nonlinear Stability. NASA Technical Memorandum 101086 (64), 1–154 (1989)

[55] Nartsissov, Y.R.: Application of a multicomponent model of convectional reaction-diffusion to description of glucose gradients in a neurovascular unit. Front. Physiol. 13, 843473 (2022). https://doi.org/10.3389/fphys.2022.843473

[56] Newell AC. Finite amplitude instabilities of partial difference equations. SIAM J. Appl. Math., 1977, 33(1): 133-160.

[57] Nordström J, Carpenter MH. Boundary and interface conditions for high-order finite-difference methods applied to the Euler and Navier-Stokes equations. J. Comput. Phys., 1999, 148(2): 621-645.

[58] Nordström J, Carpenter MH. High-order finite-difference methods, multidimensional linear problems, and curvilinear coordinates. J. Comput. Phys., 2001, 173(1): 149-174.

[59] Parsani, M., Boukharfane, R., Nolasco, I.R., Del Rey Fernández, D.C., Zampini, S., Hadri, B., Dalcin, L.: High-order accurate entropy-stable discontinuous collocated Galerkin methods with the summation-by-parts property for compressible CFD frameworks: scalable SSDC algorithms and flow solver. J. Comput. Phys. 424, 109844 (2021). https://doi.org/10.1016/j.jcp.2020.109844

[60] Parsani M, Carpenter MH, Fisher TC, Nielsen EJ. Entropy stable staggered grid discontinuous spectral collocation methods of any order for the compressible Navier-Stokes equations. SIAM J. Sci. Comput., 2016, 38(5): 3129-3162.

[61] Parsani M, Carpenter MH, Nielsen EJ. Entropy stable discontinuous interfaces coupling for the three-dimensional compressible Navier-Stokes equations. J. Comput. Phys., 2015, 290: 132-138.

[62] Parsani M, Carpenter MH, Nielsen EJ. Entropy stable wall boundary conditions for the three-dimensional compressible Navier-Stokes equations. J. Comput. Phys., 2015, 292(1): 88-113.

[63] Ranocha H, Sayyari M, Dalcin L, Parsani M, Ketcheson DI. Relaxation Runge-Kutta methods: fully-discrete explicit entropy-stable schemes for the Euler and Navier-Stokes equations. SIAM J. Sci. Comput., 2019, 42(2): 612-638.

[64] Roy CJ. Review of code and solution verification procedures for computational simulation. J. Comput. Phys., 2005, 205(1): 131-156.

[65] Serre DSystems of Conservation Laws 2: Geometric Structures, Oscillations, and Initial-Boundary Value Problems, 1999CambridgeCambridge University Press.

[66] Shear D. An analog of the Boltzmann H-theorem (a Liapunov function) for systems of coupled chemical reactions. J. Theor. Biol., 1967, 16(2): 212-228.

[67] Smoller, J.: Shock Waves and Reaction—Diffusion Equations. Springer, New York (2012). https://doi.org/10.1007/978-1-4684-0152-3

[68] Smyshlyaev A, Cerpa E, Krstic M. Boundary stabilization of a 1-D wave equation with in-domain antidamping. SIAM J. Control. Optim., 2010, 48(6): 4014-4031.

[69] Sun Z, Carrillo JA, Shu C-W. An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems. Kinetic Relat. Model., 2019, 12(4): 885-908.

[70] Svärd M, Nordström J. A stable high-order finite difference scheme for the compressible Navier-Stokes equations: no-slip wall boundary conditions. J. Comput. Phys., 2008, 227(10): 4805-4824.

[71] Svärd M, Nordström J. Review of summation-by-parts schemes for initial-boundary-value-problems. J. Comput. Phys., 2014, 268(1): 17-38.

[72] Tadmor E. Skew-selfadjoint form for systems of conservation laws. J. Math. Anal. Appl., 1984, 103(2): 428-442.

[73] Tadmor, E.: Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer. 12, 451–512 (2003). https://doi.org/10.1017/S0962492902000156

[74] Umbricht, G.F., Rubio, D., Tarzia, D.A.: Bilayer one-dimensional convection-diffusion-reaction-source problem: analytical and numerical solution. Int. J. Thermal Sci. 208, 109471 (2025). https://doi.org/10.1016/j.ijthermalsci.2024.109471

[75] Vargas-De-León, C.: Volterra-type Lyapunov functions for fractional-order epidemic systems. Commun. Nonlinear Sci. Numer. Simul. 24(1), 75–85 (2015). https://doi.org/10.1016/j.cnsns.2014.12.013

[76] Wang S, Diagne M, Qi J. Delay-adaptive predictor feedback control of reaction-advection-diffusion PDEs with a delayed distributed input. IEEE Trans. Autom. Control, 2022, 67(7): 3762-3769.

[77] Wang Q, Yang J, Zhang L. Time-periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: effect of cellular growth. Discrete Contin. Dyn. Syst. B, 2017, 22(9): 3547-3574.

[78] Whitham GB. Linear and Nonlinear Waves. John Wiley & Sons, Inc., 1999.

[79] Yamaleev, N.K., Fernandez, D.C.D.R., Lou, J., Carpenter, M.H.: Entropy stable spectral collocation schemes for the 3-D Navier-Stokes equations on dynamic unstructured grids. J. Comput. Phys. 399, 108897 (2019) https://doi.org/10.1016/j.jcp.2019.108897

[80] Yang, J., Crean, J., Hicken, J.E.: Interior penalties for summation-by-parts discretizations of linear second-order differential equations. J. Sci. Comput. 75, 1385–1414 (2018) https://doi.org/10.1007/s10915-017-0591-8

Funding

King Abdullah University of Science and Technology (KAUST)

RIGHTS & PERMISSIONS

The Author(s)

Just Accepted

This article has successfully passed peer review and final editorial review, and will soon enter typesetting, proofreading and other publishing processes. The currently displayed version is the accepted final manuscript. The officially published version will be updated with format, DOI and citation information upon launch. We recommend that you pay attention to subsequent journal notifications and preferentially cite the officially published version. Thank you for your support and cooperation.