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

doi:10.1007/s42967-025-00514-1

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (4): 1604-1652.doi: 10.1007/s42967-025-00514-1

• • 上一篇

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

Rasha Al Jahdali¹, David C. Del Rey Fernández², Lisandro Dalcin¹, Matteo Parsani^1,3

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

收稿日期:2024-10-28 修回日期:2025-04-27 接受日期:2025-06-14 出版日期:2026-08-20 发布日期:2026-09-17
通讯作者: Rasha Al Jahdali, Email: rasha.aljahdali@kaust.edu.sa E-mail:rasha.aljahdali@kaust.edu.sa
基金资助:
Open access publishing provided by King Abdullah University of Science and Technology (KAUST).

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

Rasha Al Jahdali¹, David C. Del Rey Fernández², Lisandro Dalcin¹, Matteo Parsani^1,3

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:2024-10-28 Revised:2025-04-27 Accepted:2025-06-14 Online:2026-08-20 Published:2026-09-17
Supported by:
Open access publishing provided by King Abdullah University of Science and Technology (KAUST).

摘要/Abstract

摘要： 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.

关键词: Convection-diffusion-reaction equations, Lyapunov functionals, Summation-by-parts (SBP) operators, Relaxation Runge-Kutta schemes, Fully discrete Lyapunov-consistent discretizations

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.

Key words: Convection-diffusion-reaction equations, Lyapunov functionals, Summation-by-parts (SBP) operators, Relaxation Runge-Kutta schemes, Fully discrete Lyapunov-consistent discretizations

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.

[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, D.C., 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). https://doi.org/10.1007/s42967-024-00425-7
[3] Alhumaizi, K., Henda, R., Soliman, M.: Numerical analysis of a reaction-diffusion-convection system. Comput. Chem. Eng. 27(4), 579-594 (2003). https://doi.org/10.1016/S0098-1354(02)00238-7
[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, D.R., Shankar, S., Toschi, F., Zhu, X.: Spatial population genetics with fluid flow. Rep. Prog. Phys. 85(9), 096601 (2022). https://doi.org/10.1088/1361-6633/ac8231
[6] Bertsekas, D.P.: Nonlinear programming. J. Oper. Res. Soc. 48(3), 334-334 (1997)
[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, J.C.: Numerical Methods for Ordinary Differential Equations. John Wiley & Sons, Ltd (2008). https://doi.org/10.1002/9780470753767
[13] 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
[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, 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] 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, 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. Comput. Fluids 95(22), 171-196 (2014). https://doi.org/10.1016/j.compfluid.2014.02.016
[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. 236(2), 551-569 (2007). https://doi.org/10.1016/j.jde.2007.02.002
[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, T.C., Carpenter, M.H.: High-order entropy stable finite difference schemes for nonlinear conservation laws: finite domains. J. Comput. Phys. 252(1), 518-557 (2013). https://doi.org/10.1016/j.jcp.2013.06.014
[29] Fisher, T.C., Carpenter, M.H., Nordström, J., Yamaleev, N.K.: Discretely conservative finite-difference formulations for nonlinear conservation laws in split form: theory and boundary conditions. J. Comput. Phys. 234(1), 353-375 (2013). https://doi.org/10.1016/j.jcp.2012.09.026
[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, D.A., Galeano, C.H., Mantilla, J.M.: Computational examples of reaction-convection-diffusion equations solution under the influence of fluid flow: first example. Appl. Math. Model. 36(10), 5029-5045 (2012). https://doi.org/10.1016/j.apm.2011.12.041
[32] Greenberg, J.M., Ta Tsien, L.: The effect of boundary damping for the quasilinear wave equation. J. Differ. Equ. 52(1), 66-75 (1984). https://doi.org/10.1016/0022-0396(84)90135-9
[33] 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
[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, J.M.: Equilibria of Runge-Kutta methods. Numer. Math. 58(1), 243-254 (1990). https://doi.org/10.1007/BF01385623
[36] Harten, A.: On the symmetric form of systems of conservation laws with entropy. J. Comput. Phys. 49(1), 151-164 (1983). https://doi.org/10.1016/0021-9991(83)90118-3
[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, 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
[40] Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice-Hall, Upper Saddle River, NJ (2002)
[41] Knoll, D.A.: An improved convection scheme applied to recombining divertor plasma flows. J. Comput. Phys. 142(2), 473-488 (1998). https://doi.org/10.1006/jcph.1998.5947
[42] Komornik, V.: Exact Controllability and Stabilization: the Multiplier Method, vol. 39. Wiley, Chichester (1994)
[43] Korobeinikov, A., Wake, G.C.: Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models. Appl. Math. Lett. 15(8), 955-960 (2002)
[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 44(1), 63-74 (2008). https://doi.org/10.1016/j.automatica.2007.05.012
[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. 57(9), 750-758 (2008). https://doi.org/10.1016/j.sysconle.2008.02.005
[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, P.D.: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. SIAM (1973). https://doi.org/10.1137/1.9781611970562.fm
[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). https://doi.org/10.1007/978-1-4471-0419-3
[52] Mattsson, K.: Boundary procedures for summation-by-parts operators. J. Sci. Comput. 18(1), 133-153 (2003). https://doi.org/10.1023/A:1020342429644
[53] 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
[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, A.C.: Finite amplitude instabilities of partial difference equations. SIAM J. Appl. Math. 33(1), 133-160 (1977). https://doi.org/10.1137/0133010
[57] 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
[58] 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
[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, 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
[61] 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
[62] Parsani, M., Carpenter, M.H., Nielsen, E.J.: Entropy stable wall boundary conditions for the three-dimensional compressible Navier-Stokes equations. J. Comput. Phys. 292(1), 88-113 (2015). https://doi.org/10.1016/j.jcp.2015.03.026
[63] 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), 612-638 (2019). https://doi.org/10.1137/19M1263480
[64] 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
[65] Serre, D.: Systems of Conservation Laws 2: Geometric Structures, Oscillations, and Initial-Boundary Value Problems. Cambridge University Press, Cambridge (1999)
[66] Shear, D.: An analog of the Boltzmann H-theorem (a Liapunov function) for systems of coupled chemical reactions. J. Theor. Biol. 16(2), 212-228 (1967). https://doi.org/10.1016/0022-5193(67)90005-7
[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. 48(6), 4014-4031 (2010). https://doi.org/10.1137/080742646
[69] Sun, Z., Carrillo, J.A., Shu, C.-W.: An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems. Kinetic Relat. Model. 12(4), 885-908 (2019). https://doi.org/10.3934/krm.2019033
[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. 227(10), 4805-4824 (2008). https://doi.org/10.1016/j.jcp.2007.12.028
[71] 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
[72] Tadmor, E.: Skew-selfadjoint form for systems of conservation laws. J. Math. Anal. Appl. 103(2), 428-442 (1984). https://doi.org/10.1016/0022-247X(84)90139-2
[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 67(7), 3762-3769 (2022). https://doi.org/10.1109/TAC.2021.3109512
[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 22(9), 3547-3574 (2017). https://doi.org/10.3934/dcdsb.2017179
[78] Whitham, G.B.: Linear and Nonlinear Waves. John Wiley & Sons, Inc. (1999). https://doi.org/10.1002/9781118032954
[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

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

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

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