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Local Discontinuous Galerkin Methods Coupled with Implicit Integration Factor Methods for Solving Reaction‐Cross‐Diffusion Systems

Author

Listed:
  • Na An
  • Xijun Yu
  • Chaobao Huang
  • Maochang Duan

Abstract

We present a new numerical method for solving nonlinear reaction‐diffusion systems with cross‐diffusion which are often taken as mathematical models for many applications in the biological, physical, and chemical sciences. The two‐dimensional system is discretized by the local discontinuous Galerkin (LDG) method on unstructured triangular meshes associated with the piecewise linear finite element spaces, which can derive not only numerical solutions but also approximations for fluxes at the same time comparing with most of study work up to now which has derived numerical solutions only. Considering the stability requirement for the explicit scheme with strict time step restriction (Δt=O(hmin2)), the implicit integration factor (IIF) method is employed for the temporal discretization so that the time step can be relaxed as Δt = O(hmin). Moreover, the method allows us to compute element by element and avoids solving a global system of nonlinear algebraic equations as the standard implicit schemes do, which can reduce the computational cost greatly. Numerical simulations about the system with exact solution and the Brusselator model, which is a theoretical model for a type of autocatalytic chemical reaction, are conducted to confirm the expected accuracy, efficiency, and advantages of the proposed schemes.

Suggested Citation

  • Na An & Xijun Yu & Chaobao Huang & Maochang Duan, 2016. "Local Discontinuous Galerkin Methods Coupled with Implicit Integration Factor Methods for Solving Reaction‐Cross‐Diffusion Systems," Discrete Dynamics in Nature and Society, John Wiley & Sons, vol. 2016(1).
    • Handle: RePEc:wly:jnddns:v:2016:y:2016:i:1:n:5345032
    DOI: 10.1155/2016/5345032
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    1. Hezi Yizhaq & Boris A Portnov & Ehud Meron, 2004. "A Mathematical Model of Segregation Patterns in Residential Neighbourhoods," Environment and Planning A, , vol. 36(1), pages 149-172, January.
    2. Gambino, G. & Lombardo, M.C. & Sammartino, M., 2012. "Turing instability and traveling fronts for a nonlinear reaction–diffusion system with cross-diffusion," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(6), pages 1112-1132.
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