## Taiwanese Journal of Mathematics

### LDG Methods for Reaction-diffusion Systems with Application of Krylov Implicit Integration Factor Methods

#### Abstract

In this paper, we present an efficient fully-discrete local discontinuous Galerkin (LDG) method for nonlinear reaction-diffusion systems, which are often used as mathematical models for many physical and biological applications. We can derive numerical approximations not only for solutions but also for their gradients at the same time, while most of methods derive numerical solutions only. And due to the strict time-step restriction ($\Delta t = O(h^2_{\min})$) of explicit schemes for stability, we introduce the implicit integration factor (IIF) method based on Krylov subspace approximation, in which the time step can be taken as $\Delta t = O(h_{\min})$. Moreover, the method allows us to compute element by element and avoid solving a global system of nonlinear algebraic equations as the standard implicit schemes do, which can reduce the computational cost greatly. Numerical experiments about the reaction-diffusion equations with exact solutions and the well-studied Schnakenberg system are conducted to illustrate the accuracy, capability and advantages of the method.

#### Article information

Source
Taiwanese J. Math., Volume 23, Number 3 (2019), 727-749.

Dates
Revised: 4 July 2018
Accepted: 9 September 2018
First available in Project Euclid: 26 September 2018

https://projecteuclid.org/euclid.twjm/1537927428

Digital Object Identifier
doi:10.11650/tjm/180902

Mathematical Reviews number (MathSciNet)
MR3952249

Zentralblatt MATH identifier
07068572

Subjects
Primary: 35K57: Reaction-diffusion equations

#### Citation

An, Na; Huang, Chaobao; Yu, Xijun. LDG Methods for Reaction-diffusion Systems with Application of Krylov Implicit Integration Factor Methods. Taiwanese J. Math. 23 (2019), no. 3, 727--749. doi:10.11650/tjm/180902. https://projecteuclid.org/euclid.twjm/1537927428

#### References

• F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys. 131 (1997), no. 2, 267–279.
• P. Castillo, B. Cockburn, I. Perugia and D. Schötzau, An a priori error analysis of the local discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal. 38 (2000), no. 5, 1676–1706.
• Z. Chen, B. Cockburn, C. Gardner and J. W. Jerome, Quantum hydrodynamic simulation of hysteresis in the resonant tunneling diode, J. Comput. Phys. 117 (1995), no. 2, 274–280.
• Z. Chen, B. Cockburn, J. W. Jerome and C.-W. Shu, Mixed-RKDG finite element methods for the 2-D hydrodynamic model for semiconductor device simulation, VLSI Design 3 (1995), no. 2, 145–158.
• S. Chen and Y.-T. Zhang, Krylov implicit integration factor methods for spatial discretization on high dimensional unstructured meshes: application to discontinuous Galerkin methods, J. Comput. Phys. 230 (2011), no. 11, 4336–4352.
• Y. Cheng and C.-W. Shu, A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives, Math. Comp. 77 (2008), no. 262, 699–730.
• B. Cockburn, G. Kanschat, I. Perugia and D. Schötzau, Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids, SIAM J. Numer. Anal. 39 (2002), no. 1, 264–285.
• B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal. 35 (1998), no. 6, 2440–2463.
• A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik 12 (1972), no. 1, 30–39.
• B. C. Goodwin and L. E. H. Trainor, Tip and whorl morphogenesis in Acetabularia by calcium-regulated strain fields, J. Theoret. Biol. 117 (1985), no. 1, 79–106.
• P. Gray and S. K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Isolas and other forms of multistability, Chem. Eng. Sci. 38 (1983), no. 1, 29–43.
• R. Guo, Y. Xia and Y. Xu, Semi-implicit spectral deferred correction methods for highly nonlinear partial differential equations, J. Comput. Phys. 338 (2017), 269–284.
• A. L. Hanhart, M. K. Gobbert and L. T. Izu, A memory-efficient finite element method for systems of reaction-diffusion equations with non-smooth forcing, J. Comput. Appl. Math. 169 (2004), no. 2, 431–458.
• W. Hundsdorfer and J. Verwer, Numerical solution of time-dependent advection-diffusion-reaction equations, Springer Series in Computational Mathematics 33, Springer-Verlag, Berlin, 2003.
• P. Jamet, Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain, SIAM J. Numer. Anal. 15 (1978), no. 5, 912–928.
• G. S. Jiang and C.-W. Shu, On a cell entropy inequality for discontinuous Galerkin methods, Math. Comp. 62 (1994), no. 206, 531–538.
• C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge, 1987.
• A. Kassam and L. N. Trefethen, Fourth-order time-stepping for stiff PDEs, SIAM J. Sci. Comput. 26 (2005), no. 4, 1214–1233.
• I. Lengyel and I. R. Epstein, Modeling of turing structures in the chlorite-iodide-malonic acid-starch reaction system, Science 251 (1991), no. 4994, 650–652.
• B. Q. Li, Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer, Computational Fluid and Solid Mechanics, Springer-Verlag London, London, 2006.
• H. Liu and J. Yan, The direct discontinuous Galerkin (DDG) methods for diffusion problems, SIAM J. Numer. Anal. 47 (2009), no. 1, 675–698.
• A. Madzvamuse, Time-stepping schemes for moving grid finite elements applied to reaction-diffusion systems on fixed and growing domains, J. Comput. Phys. 214 (2006), no. 1, 239–263.
• A. Madzvamuse, A. J. Wathen and P. K. Maini, A moving grid finite element method applied to a model biological pattern generator, J. Comput. Phys. 190 (2003), no. 2, 478–500.
• Q. Nie, Y.-T. Zhang and R. Zhao, Efficient semi-implicit schemes for stiff systems, J. Comput. Phys. 214 (2006), no. 2, 512–537.
• W. H. Reed and T. R. Hill, Triangular mesh methods for the neutron transport equation, Los Alamos Scientific Laboratory report LA-UR-73-479, Los Alamos, NM, 1973.
• D. L. Ropp, J. N. Shadid and C. C. Ober, Studies of the accuracy of time integration methods for reaction-diffusion equations, J. Comput. Phys. 194 (2004), no. 2, 544–574.
• S. J. Ruuth, Implicit-explicit methods for reaction-diffusion problems in pattern formation, J. Math. Biol. 34 (1995), no. 2, 148–176.
• J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour, J. Theoret. Biol. 81 (1979), no. 3, 389–400.
• A. M. Soane, M. K. Gobbert and T. I. Seidman, Numerical exploration of a system of reaction-diffusion equations with internal and transient layers, Nonlinear Anal. Real World Appl. 6 (2005), no. 5, 914–934.
• A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B 237 (1952), no. 641, 37–72.
• Y. Xu and C.-W. Shu, Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations, Comput. Methods Appl. Mech. Engrg. 196 (2007), no. 37-40, 3805–3822.
• R. Zhang, X. Yu, J. Zhu and A. F. D. Loula, Direct discontinuous Galerkin method for nonlinear reaction-diffusion systems in pattern formation, Appl. Math. Model. 38 (2014), no. 5-6, 1612–1621.
• J. Zhu, Y.-T. Zhang, S. A. Newman and M. Alber, Application of discontinuous Galerkin methods for reaction-diffusion systems in developmental biology, J. Sci. Comput. 40 (2009), no. 1-3, 391–418.