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 AJCM  Vol.3 No.2 , June 2013
Computational Studies of Reaction-Diffusion Systems by Nonlinear Galerkin Method
Abstract: This article deals with the computational study of the nonlinear Galerkin method, which is the extension of commonly known Faedo-Galerkin method. The weak formulation of the method is derived and applied to the particular Scott-Wang-Showalter reaction-diffusion model concerning the problem of combustion of hydrocarbon gases. The proof of convergence of the method based on the method of compactness is introduced. Presented results of numerical simulations are composed of the computational study, where the nonlinear Galerkin method and Faedo-Galerkin method are compared for the problem with analytical solution and the numerical results of the Scott-Wang-Showalter model in 1D.
Cite this paper: M. Kolář, "Computational Studies of Reaction-Diffusion Systems by Nonlinear Galerkin Method," American Journal of Computational Mathematics, Vol. 3 No. 2, 2013, pp. 137-146. doi: 10.4236/ajcm.2013.32022.
References

[1]   M. Marion and R. Temam, “Nonlinear Galerkin Methods,” SIAM Journal on Numerical Analysis, Vol. 26, No. 5, 1989, pp. 1139-1157. doi:10.1137/0726063

[2]   J. Sembera and M. Benes, “Nonlinear Galerkin Method for Reaction-Diffusion Systems Admitting Invariant Regions,” Journal of Computational and Applied Mathematics, Vol. 136, No. 1-2, 2001, pp. 163-176. doi:10.1016/S0377-0427(00)00582-3

[3]   J. Mach, “Application of the Nonlinear Galerkin FEM Method to the Solution of the Reaction Diffusion Equations,” Journal of Math-for-Industry, Vol. 3, 2011, pp. 41-51.

[4]   M. Kolár, “Mathematical Modelling and Numerical Simulations of Reaction-Diffusion Processes,” Diploma Thesis, Department of Mathematics FNSPE CTU, Prague, 2012.

[5]   A. Debussche and M. Marion, “On the Construction of Families of Approximate Inertial Manifolds,” Journal of Differential Equations, Vol. 100, No. 1, 1992, pp. 173-201. doi:10.1016/0022-0396(92)90131-6

[6]   S. K. Scott, J. Wang and K. Showalter, “Modelling Studies of Spiral Waves and Target Patterns in Premixed Flames,” Journal of the Chemical Society, Faraday Transactions, Vol. 93, No. 9, 1997, pp. 1733-1739. doi:10.1039/a608474e

[7]   V. Tomica, “Reaction-Diffusion Equations in Combustion,” Proceedings of Czech Japanese Seminar in Applied Mathematics, Prague, 30 August-4 September 2010, pp. 84-93.

[8]   R. Temam, “Infinite-Dimensional Dynamical Systems in Mechanics and Physics,” Springer, Berlin, 1997.

[9]   J. L. Lions, “Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires,” Dunod, Paris, 1969.

 
 
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