Local Discontinuous Galerkin Methods Coupled with Implicit Integration Factor Methods for Solving Reaction-Cross-Diffusion Systems
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...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Wiley
2016-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2016/5345032 |
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| author | Na An Xijun Yu Chaobao Huang Maochang Duan |
| author_facet | Na An Xijun Yu Chaobao Huang Maochang Duan |
| author_sort | Na An |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-c4041855dc7c4dd49542bda46eaa99fe |
| institution | OA Journals |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2016-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-c4041855dc7c4dd49542bda46eaa99fe2025-08-20T02:18:57ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2016-01-01201610.1155/2016/53450325345032Local Discontinuous Galerkin Methods Coupled with Implicit Integration Factor Methods for Solving Reaction-Cross-Diffusion SystemsNa An0Xijun Yu1Chaobao Huang2Maochang Duan3Graduate School of China Academy of Engineering Physics, Beijing 100088, ChinaLaboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, ChinaGraduate School of China Academy of Engineering Physics, Beijing 100088, ChinaGraduate School of China Academy of Engineering Physics, Beijing 100088, ChinaWe 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.http://dx.doi.org/10.1155/2016/5345032 |
| spellingShingle | Na An Xijun Yu Chaobao Huang Maochang Duan Local Discontinuous Galerkin Methods Coupled with Implicit Integration Factor Methods for Solving Reaction-Cross-Diffusion Systems Discrete Dynamics in Nature and Society |
| title | Local Discontinuous Galerkin Methods Coupled with Implicit Integration Factor Methods for Solving Reaction-Cross-Diffusion Systems |
| title_full | Local Discontinuous Galerkin Methods Coupled with Implicit Integration Factor Methods for Solving Reaction-Cross-Diffusion Systems |
| title_fullStr | Local Discontinuous Galerkin Methods Coupled with Implicit Integration Factor Methods for Solving Reaction-Cross-Diffusion Systems |
| title_full_unstemmed | Local Discontinuous Galerkin Methods Coupled with Implicit Integration Factor Methods for Solving Reaction-Cross-Diffusion Systems |
| title_short | Local Discontinuous Galerkin Methods Coupled with Implicit Integration Factor Methods for Solving Reaction-Cross-Diffusion Systems |
| title_sort | local discontinuous galerkin methods coupled with implicit integration factor methods for solving reaction cross diffusion systems |
| url | http://dx.doi.org/10.1155/2016/5345032 |
| work_keys_str_mv | AT naan localdiscontinuousgalerkinmethodscoupledwithimplicitintegrationfactormethodsforsolvingreactioncrossdiffusionsystems AT xijunyu localdiscontinuousgalerkinmethodscoupledwithimplicitintegrationfactormethodsforsolvingreactioncrossdiffusionsystems AT chaobaohuang localdiscontinuousgalerkinmethodscoupledwithimplicitintegrationfactormethodsforsolvingreactioncrossdiffusionsystems AT maochangduan localdiscontinuousgalerkinmethodscoupledwithimplicitintegrationfactormethodsforsolvingreactioncrossdiffusionsystems |