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On the stability of finite difference schemes for nonlinear reaction-diffusion systems

Nathan Muyinda (UGent) , Bernard De Baets (UGent) and Shodhan Rao (UGent)
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Abstract
We identify sufficient conditions for the stability of some well-known finite difference schemes for the solution of the multivariable reaction-diffusion equations that model chemical reaction networks. Since the equations are mainly nonlinear, these conditions are obtained through local linearization. A recurrent condition is that the Jacobian matrix of the reaction part evaluated at some positive unknown solution is either D-semi-stable or semi-stable. We demonstrate that for a single reversible chemical reaction whose kinetics are monotone, the Jacobian matrix is D-semi-stable and therefore such schemes are guaranteed to work well.
Keywords
EQUATIONS, CONVERGENCE

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Chicago
Muyinda, Nathan, Bernard De Baets, and Shodhan Rao. 2018. “On the Stability of Finite Difference Schemes for Nonlinear Reaction-diffusion Systems.” In AIP Conference Proceedings. Vol. 1978. Melvillle, NY, USA: American Institute of Physics (AIP).
APA
Muyinda, N., De Baets, B., & Rao, S. (2018). On the stability of finite difference schemes for nonlinear reaction-diffusion systems. AIP Conference Proceedings (Vol. 1978). Presented at the International conference on Numerical Analysis and Applied Mathematics (ICNAAM 2017), Melvillle, NY, USA: American Institute of Physics (AIP).
Vancouver
1.
Muyinda N, De Baets B, Rao S. On the stability of finite difference schemes for nonlinear reaction-diffusion systems. AIP Conference Proceedings. Melvillle, NY, USA: American Institute of Physics (AIP); 2018.
MLA
Muyinda, Nathan, Bernard De Baets, and Shodhan Rao. “On the Stability of Finite Difference Schemes for Nonlinear Reaction-diffusion Systems.” AIP Conference Proceedings. Vol. 1978. Melvillle, NY, USA: American Institute of Physics (AIP), 2018. Print.
@inproceedings{8569716,
  abstract     = {We identify sufficient conditions for the stability of some well-known finite difference schemes for the solution of the multivariable reaction-diffusion equations that model chemical reaction networks. Since the equations are mainly nonlinear, these conditions are obtained through local linearization. A recurrent condition is that the Jacobian matrix of the reaction part evaluated at some positive unknown solution is either D-semi-stable or semi-stable. We demonstrate that for a single reversible chemical reaction whose kinetics are monotone, the Jacobian matrix is D-semi-stable and therefore such schemes are guaranteed to work well.},
  articleno    = {470004},
  author       = {Muyinda, Nathan and De Baets, Bernard and Rao, Shodhan},
  booktitle    = {AIP Conference Proceedings},
  isbn         = {9780735416901},
  keyword      = {EQUATIONS,CONVERGENCE},
  language     = {eng},
  location     = {Thessaloniki, Greece},
  pages        = {4},
  publisher    = {American Institute of Physics (AIP)},
  title        = {On the stability of finite difference schemes for nonlinear reaction-diffusion systems},
  url          = {http://dx.doi.org/10.1063/1.5044074},
  volume       = {1978},
  year         = {2018},
}

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