ORIGINAL PAPERS

Fully-Discrete Lyapunov Consistent Discretizations for Parabolic Reaction-Diffusion Equations with r Species

  • Rasha Al Jahdali ,
  • David C. Del Rey Fernández ,
  • Lisandro Dalcin ,
  • Mohammed Sayyari ,
  • Peter Markowich ,
  • Matteo Parsani
Expand
  • 1. Computer Electrical and Mathematical Science and Engineering Division (CEMSE), Extreme Computing Research Center (ECRC), 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: 2023-11-26

  Revised date: 2024-04-01

  Online published: 2026-02-11

Supported by

The work described in this paper was supported by King Abdullah University of Science and Technology through the grant number BAS/1/1663-01-01.

Abstract

Reaction-diffusion equations model various biological, physical, sociological, and environmental phenomena. Often, numerical simulations are used to understand and discover the dynamics of such systems. Following the extension of the nonlinear Lyapunov theory applied to some class of reaction-diffusion partial differential equations (PDEs), we develop the first fully discrete Lyapunov discretizations that are consistent with the stability properties of the continuous parabolic reaction-diffusion models. The proposed framework provides a systematic procedure to develop fully discrete schemes of arbitrary order in space and time for solving a broad class of equations equipped with a Lyapunov functional. The new schemes are applied to solve systems of PDEs, which arise in epidemiology and oncolytic M1 virotherapy. The new computational framework provides physically consistent and accurate results without exhibiting scheme-dependent instabilities and converging to unphysical solutions. The proposed approach represents a capstone for developing efficient, robust, and predictive technologies for simulating complex phenomena.

Cite this article

Rasha Al Jahdali , David C. Del Rey Fernández , Lisandro Dalcin , Mohammed Sayyari , Peter Markowich , Matteo Parsani . Fully-Discrete Lyapunov Consistent Discretizations for Parabolic Reaction-Diffusion Equations with r Species[J]. Communications on Applied Mathematics and Computation, 2026 , 8(1) : 195 -231 . DOI: 10.1007/s42967-024-00425-7

References

[1] Abdelmalek, S., Bendoukha, S.: Global asymptotic stability for a SEI reaction-diffusion model of infectious diseases with immigration. Int. J. Biomath. 11(03), 1850044 (2018). https://doi.org/10.1142/S1793524518500444
[2] 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.camwa.2022.05.006
[3] Al Jahdali, R., Fernández, D., Dalcin, L., Sayyari, M., Markowich, P., Parsani, M.: Reproducibility repository for “Fully discrete Lyapunov consistent discretizations of any order for parabolic reaction-diffusion equations with r species”. (2023). https://github.com/aanslab-papers/2023-fully-discrete-lyapunov-consistent-discretizations
[4] Anosov, D.V., Aranson, S.K., Arnold, V.I., Bronshtein, I.U., Il’yashenko, Y.S., Grines, V.Z.: Ordinary Differential Equations and Smooth Dynamical Systems. Springer, Berlin, Heidelberg (1997)
[5] Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742-760 (1982). https://doi.org/10.1137/0719052
[6] Biazar, J., Mehrlatifan, M.B.: A compact finite difference scheme for reaction-convection-diffusion equation. Chiang Mai J. Sci. 45(3), 1559-1568 (2018). https://doi.org/10.1186/s13662-018-1731-7
[7] Brezzi, F., Ushiki, S., Fujii, H.: “Real” and “ghost” bifurcation dynamics in difference schemes for ODEs. In: Numerical Methods for Bifurcation Problems, pp. 79-104. Springer, Basel (1984)
[8] Bueno-Orovio, A., Perez-Garcia, V.M., Fenton, F.H.: Spectral methods for partial differential equations in irregular domains: the spectral smoothed boundary method. SIAM J. Sci. Comput. 28(3), 886-900 (2006). https://doi.org/10.1137/040607575
[9] 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
[10] Burton, T.A.: Stability by Fixed Point Theory for Functional Differential Equations. Courier Corporation, Chicago (2013)
[11] Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. John Wiley & Sons Ltd (2008)
[12] Cai, Y., Chi, D., Liu, W., Wang, W.: Stationary patterns of a cross-diffusion epidemic model. Abstr. Appl. Anal. (2013). https://doi.org/10.1155/2013/852698
[13] 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
[14] Carpenter, M.H., Fisher, T.C., Nielsen, E.J., Frankel, S.H.: Entropy stable spectral collocation schemes for the Navier-Stokes equations: discontinuous interfaces. SIAM J. Sci. Comput. 36(5), 835-867 (2014). https://doi.org/10.1137/130932193
[15] Carpenter, M.H., Nordström, J., Gottlieb, D.: A stable and conservative interface treatment of arbitrary spatial accuracy. J. Comput. Phys. 148(2), 341-365 (1999). https://doi.org/10.1006/jcph.1998.6114
[16] Chernyshenko, A.Y., Olshanskii, M.A.: An adaptive octree finite element method for PDEs posed on surfaces. Comput. Methods Appl. Mech. Eng. 291, 146-172 (2015). https://doi.org/10.1016/j.cma.2015.03.025
[17] Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35(6), 2440-2463 (1998). https://doi.org/10.1137/S0036142997316712
[18] Curtain, R., Zwart, H.: Introduction to Infinite-Dimensional Systems Theory: a State-Space Approach. Springer, New York (2020)
[19] David, A.: The simulations driving the world’s response to COVID-19: how epidemiologists rushed to model the coronavirus pandemic. Nature 580(7803), 316-318 (2020)
[20] Del Rey Fernández, D.C., Carpenter, M.H., Dalcin, L., Zampini, S., Parsani, M.: Entropy stable \begin{document}$ h/p $\end{document}-nonconforming discretization with the summation-by-parts property for the compressible Euler and Navier-Stokes equations. SN Partial Differential Equations and Applications 1, 9 (2020). https://doi.org/10.1007/s42985-020-00009-z
[21] Del Rey Fernández, D.C., Hicken, J.E., Zingg, D.W.: Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations. Computers & Fluids 95(22), 171-196 (2014). https://doi.org/10.1016/j.compfluid.2014.02.016
[22] Elaiw, A.M., Hobiny, A.D., Al Agha, A.D.: Global dynamics of reaction-diffusion oncolytic M1 virotherapy with immune response. Appl. Math. Comput. 367, 124758 (2020). https://doi.org/10.1016/j.amc.2019.124758
[23] Gatenby, R.A., Gawlinski, E.T.: A reaction-diffusion model of cancer invasion. Can. Res. 56(24), 5745-5753 (1996)
[24] Greenberg, J.M., Ta Tsien, L.: The effect of boundary damping for the quasilinear wave equation. J. Differential Equations 52(1), 66-75 (1984). https://doi.org/10.1016/0022-0396(84)90135-9
[25] Griffiths, D., Sweby, P., Yee, H.C.: On spurious asymptotic numerical solutions of explicit Runge-Kutta methods. IMA J. Numer. Anal. 12(3), 319-338 (1992). https://doi.org/10.1093/imanum/12.3.319
[26] Gustafsson, B., Kreiss, H.O., Oliger, J.: Time-Dependent Problems and Difference Methods. Pure and Applied Mathematics: a Wiley Series of Texts, Monographs and Tracts. John Wiley & Sons Ltd (2013)
[27] Haddad, W.M., Chellaboina, V.: Nonlinear Dynamical Systems and Control: a Lyapunov-based Approach. Princeton University Press, Princeton, New Jersey (2008)
[28] Hairer, E., Iserles, A., Sanz-Serna, J.M.: Equilibria of Runge-Kutta methods. Numer. Math. 58(1), 243-254 (1990). https://doi.org/10.1007/BF01385623
[29] Haushofer, J., Metcalf, C.J.E.: Which interventions work best in a pandemic? Science 368(6495), 1063-1065 (2020). https://doi.org/10.1126/science.abb6144
[30] Heidari, M., Ghovatmand, M., Skandari, M.N., Baleanu, D.: Numerical solution of reaction-diffusion equations with convergence analysis. Journal of Nonlinear Mathematical Physics, 30, 384-399 (2022). https://doi.org/10.1007/s44198-022-00086-1
[31] 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
[32] Ketcheson, D.I.: Relaxation Runge-Kutta methods: conservation and stability for inner-product norms. SIAM J. Numer. Anal. 57(6), 2850-2870 (2019). https://doi.org/10.1137/19M1263662
[33] Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice-Hall, Upper Saddle River, NJ (2002)
[34] Kolev, M.K., Koleva, M.N., Vulkov, L.G.: An unconditional positivity-preserving difference scheme for models of cancer migration and invasion. Mathematics 10(1), 131 (2022). https://doi.org/10.3390/math10010131
[35] Komornik, V.: Exact Controllability and Stabilization: the Multiplier Method. John and Wiley, Chichester (1994)
[36] 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. Academic Press (1974)
[37] Krstic, M., Guo, B.-Z., Balogh, A., Smyshlyaev, A.: Output-feedback stabilization of an unstable wave equation. Automatica 44(1), 63-74 (2008). https://doi.org/10.1016/j.automatica.2007.05.012
[38] Krstic, M., Smyshlyaev, A.: Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays. Systems & Control Letters 57(9), 750-758 (2008). https://doi.org/10.1016/j.sysconle.2008.02.005
[39] LaSalle, J.P., Artstein, Z.: The Stability of Dynamical Systems. CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (1976)
[40] LaSalle, J.P., Lefschetz, S.: Stability by Lyapunov’s Direct Method: with Applications. Mathematics in Science and Engineering: a Series of Monographs and Textbooks. Academic Press, New York (1961)
[41] Liu, Y., Cai, J., Liu, W., Lin, Y., Guo, L., Liu, X., Qin, Z., Xu, C., Zhang, Y., Su, X., Deng, K., Yan, G., Liang, J.: Intravenous injection of the oncolytic virus M1 awakens antitumor T cells and overcomes resistance to checkpoint blockade. Cell Death & Disease 11(12), 1062 (2020). https://doi.org/10.1038/s41419-020-03285-0
[42] Lyapunov, A.M.: The general problem of the stability of motion (In Russian). Ph.D. thesis, University of Kharkov (1892)
[43] Lyapunov, A.M.: The general problem of the stability of motion. Int. J. Control 55(3), 531-534 (1992). https://doi.org/10.1080/00207179208934253
[44] Macías-Díaz, J.: On the numerical and structural properties of a logarithmic scheme for diffusion-reaction equations. Appl. Numer. Math. 140, 104-114 (2019). https://doi.org/10.1016/j.apnum.2019.01.015
[45] Macías-Díaz, J.E., Puri, A.: An explicit positivity-preserving finite-difference scheme for the classical Fisher-Kolmogorov-Petrovsky-Piscounov equation. Appl. Math. Comput. 218(9), 5829-5837 (2012). https://doi.org/10.1016/j.amc.2011.11.064
[46] Mañosa, V.: Periodic travelling waves in nonlinear reaction-diffusion equations via multiple Hopf bifurcation. Chaos, Solitons & Fractals 18(2), 241-257 (2003). https://doi.org/10.1016/S0960-0779(02)00645-8
[47] Marelli, G., Howells, A., Lemoine, N.R., Wang, Y.: Oncolytic viral therapy and the immune system: a double-edged sword against cancer. Front. Immunol. 9, 866 (2018). https://doi.org/10.3389/fimmu.2018.00866
[48] Mattsson, K.: Boundary procedures for summation-by-parts operators. J. Sci. Comput. 18(1), 133-153 (2003). https://doi.org/10.1023/A:1020342429644
[49] Mattsson, K., Ham, F., Iaccarino, G.: Stable boundary treatment for the wave equation on second-order form. J. Sci. Comput. 41(3), 366-383 (2009). https://doi.org/10.1007/s10915-009-9305-1
[50] Newell, A.C.: Finite amplitude instabilities of partial difference equations. SIAM J. Appl. Math. 33(1), 133-160 (1977). https://doi.org/10.1137/0133010
[51] Nordström, J., Carpenter, M.H.: Boundary and interface conditions for high-order finite-difference methods applied to the Euler and Navier-Stokes equations. J. Comput. Phys. 148(2), 621-645 (1999). https://doi.org/10.1006/jcph.1998.6133
[52] Nordström, J., Carpenter, M.H.: High-order finite-difference methods, multidimensional linear problems, and curvilinear coordinates. J. Comput. Phys. 173(1), 149-174 (2001). https://doi.org/10.1006/jcph.2001.6864
[53] 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
[54] Parsani, M., Carpenter, M.H., Fisher, T.C., Nielsen, E.J.: Entropy stable staggered grid discontinuous spectral collocation methods of any order for the compressible Navier-Stokes equations. SIAM J. Sci. Comput. 38(5), 3129-3162 (2016). https://doi.org/10.1137/15M1043510
[55] Parsani, M., Carpenter, M.H., Nielsen, E.J.: Entropy stable discontinuous interfaces coupling for the three-dimensional compressible Navier-Stokes equations. J. Comput. Phys. 290, 132-138 (2015). https://doi.org/10.1016/j.jcp.2015.02.042
[56] Ranocha, H., Sayyari, M., Dalcin, L., Parsani, M., Ketcheson, D.I.: Relaxation Runge-Kutta methods: fully-discrete explicit entropy-stable schemes for the Euler and Navier-Stokes equations. SIAM J. Sci. Comput. 42(2), A585-C68 (2020). https://doi.org/10.1137/19M1263480
[57] Reitz, R.D.: A study of numerical methods for reaction-diffusion equations. SIAM J. Sci. Stat. Comput. 2(1), 95-106 (1981). https://doi.org/10.1137/0902008
[58] Roy, C.J.: Review of code and solution verification procedures for computational simulation. J. Comput. Phys. 205(1), 131-156 (2005). https://doi.org/10.1016/j.jcp.2004.10.036
[59] Sigdel, R.P., McCluskey, C.C.: Global stability for an SEI model of infectious disease with immigration. Appl. Math. Comput. 243, 684-689 (2014). https://doi.org/10.1016/j.amc.2014.06.020
[60] Smyshlyaev, A., Cerpa, E., Krstic, M.: Boundary stabilization of a 1-D wave equation with in-domain antidamping. SIAM J. Control. Optim. 48(6), 4014-4031 (2010). https://doi.org/10.1137/080742646
[61] Svärd, M., Carpenter, M.H., Nordström, J.: A stable high-order finite difference scheme for the compressible Navier-Stokes equations, far-field boundary conditions. J. Comput. Phys. 225(1), 1020-1038 (2007). https://doi.org/10.1016/j.jcp.2007.01.023
[62] Svärd, M., Nordström, J.: Review of summation-by-parts schemes for initial-boundary-value-problems. J. Comput. Phys. 268(1), 17-38 (2014). https://doi.org/10.1016/j.jcp.2014.02.031
[63] Thielscher, A., Antunes, A., Saturnino, G.B.: Field modeling for transcranial magnetic stimulation: a useful tool to understand the physiological effects of TMS? In: 2015 37th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), pp. 222-225 (2015). https://doi.org/10.1109/EMBC.2015.7318340
[64] Thomas, P.D., Lombard, C.K.: Geometric conservation law and its application to flow computations on moving grids. AIAA J. 17(10), 1030-1037 (1979). https://doi.org/10.2514/3.61273
[65] Toubaei, S., Garshasbi, M., Jalalvand, M.: A numerical treatment of a reaction-diffusion model of spatial pattern in the embryo. Computational Methods for Differential Equations 4(2), 116-127 (2016)
[66] Wang, W., Cai, Y., Wu, M., Wang, K., Li, Z.: Complex dynamics of a reaction-diffusion epidemic model. Nonlinear Anal. Real World Appl. 13(5), 2240-2258 (2012). https://doi.org/10.1016/j.nonrwa.2012.01.018
[67] Wang, W., Zhao, X.-Q.: Basic reproduction numbers for reaction-diffusion epidemic models. SIAM J. Appl. Dyn. Syst. 11(4), 1652-1673 (2012). https://doi.org/10.1137/120872942
[68] Xiao, X., Wang, K., Feng, X.: A lifted local Galerkin method for solving the reaction-diffusion equations on implicit surfaces. Comput. Phys. Commun. 231, 107-113 (2018). https://doi.org/10.1016/j.cpc.2018.04.023
[69] Zhang, J., Liu, Y., Tan, J., Zhang, Y., Wong, C.-W., Lin, Z., Liu, X., Sander, M., Yang, X., Liang, L., Song, D., Dan, J., Zhou, Y., Cai, J., Lin, Y., Liang, J., Hu, J., Yan, G., Zhu, W.: Necroptotic virotherapy of oncolytic alphavirus M1 cooperated with Doxorubicin displays promising therapeutic efficacy in TNBC. Oncogene 40(29), 4783-4795 (2021). https://doi.org/10.1038/s41388-021-01869-4
Options
Outlines

/