Dynamics of curved reaction fronts under a single-equation model

Bhanot, Rajeev Prakash ; Strunin, Dmitry

ANZIAM Journal, Tome 59 (2018), / Harvested from Australian Mathematical Society

Résumé

Fronts of reaction in certain systems (such as so-called solid flames) are modelled by a high-order nonlinear partial differential equation, which we analyse numerically. Previously, Strunin [IMA J. Appl. Math. 63:163--177, 1999] obtained stable spinning solutions of the equation using the Galerkin method. Here we use a more sophisticated and arguably more powerful method, namely the one-dimensional radial basis function method, to study the equation further. As an initial step, we elaborate the numerical code and tested it by reproducing the spinning regimes for a range of initial conditions. In a new series of experiments, we find a regime where the front is shaped as a pair of kinks spinning in a stable joint formation. The settled character of this regime is demonstrated. References D. V. Strunin. Autosoliton model of the spinning fronts of reaction. IMA J. Appl. Math. 63:163–177, 1999. doi:10.1093/imamat/63.2.163 D. V. Strunin. Phase equation with nonlinear excitation for nonlocally coupled oscillators. Physica D 238:1909–1916, 2009 doi:10.1016/j.physd.2009.06.022 D. V. Strunin and M. G. Mohammed. Range of validity and intermittent dynamics of the phase of oscillators with nonlinear self-excitation. Commun. Nonlinear Sci. 29:128–147, 2015. doi:10.1016/j.cnsns.2015.04.024 F. J. Mohammed, D. Ngo-Cong, D. V. Strunin, N. Mai-Duy and T. Tran-Cong. Modelling dispersion in laminar and turbulent flows in an open channel based on centre manifolds using 1D-IRBFN method. Appl. Math. Model. 38:3672–3691, 2014. doi:10.1016/j.apm.2013.12.007 N. Mai-Duy and T. Tran-Cong. Numerical solution of Navier–Stokes equations using multiquadric radial basis function networks. Int. J. Numer. Meth. Fluids 37:65–86, 2001. doi:10.1002/fld.165 N. Mai-Duy and R. I. Tanner. A collocation method based on one-dimensional RBF interpolation scheme for solving PDEs. Int. J. Numer. Meth. Heat Fluid 17:165–186, 2007. doi:10.1108/09615530710723948 D. Ngo-Cong, N. Mai-Duy, W. Karunasena and T. Tran-Cong. Local moving least square- one-dimensional IRBFN technique for incompressible viscous flows. Int. J. Numer. Meth. Fluids 70:1443–1474, 2012. doi:10.1002/fld.3640 D. Ngo-Cong, N. Mai-Duy, W. Karunasena and T. Tran-Cong. Free vibration analysis of laminated composite plates based on FSDT using one-dimensional IRBFN Method. Comput. Struct. 89:1–13, 2011. doi:10.1016/j.compstruc.2010.07.012 D. Ngo-Cong, N. Mai-Duy, W. Karunasena and T. Tran-Cong. A numerical procedure based on 1D-IRBFN and local MLS-1D-IRBFN methods for fluid-structure interaction analysis. Comput. Model. Eng. Sci. 83:459–498, 2012. doi:10.3970/cmes.2012.083.459 S. Haykin. Neural networks: A comprehensive foundation. Prentice-Hall, 1999. G. E. Fasshauer. Meshfree approximation methods with Matlab. Interdisciplinary mathematical Sciences, Vol. 6. World Scientific, 2007. doi:10.1142/6437 E. J. Kansa. Multiquadrics–-a scattered data approximation scheme with applications to computational fluid-dynamics–-I. Surface approximations partial derivative estimates. Comput, Math. Appl. 19:127–145, 1990. doi:10.1016/0898-1221(90)90270-T N. Mai-Duy and T. Tran-Cong. Numerical solution of differential equations using multiquadric radial basis function networks. Neural Net. 14:185–199, 2001. doi:10.1016/S0893-6080(00)00095-2 N. Mai-Duy and T. Tran-Cong. Approximation of function and its derivatives using radial basis function networks. Appl. Math. Model. 27:197–220, 2003. doi:10.1016/S0307-904X(02)00101-4 N. Mai-Duy and R. I. Tanner. Solving high order partial differential equations with radial basis function networks. Int. J. Numer. Meth. Eng. 63:1636–1654, 2005. doi:10.1002/nme.1332 N. Mai-Duy, H. See and T. Tran-Cong. An integral-collocation-based fictitious-domain technique for solving elliptic problems. Commun. Numer. Meth. Eng. 24:1291–1314, 2008. doi:10.1002/cnm.1033 N. Mai-Duy and T. Tran-Cong. A multidomain integrated-radial-basis-function collocation method for elliptic problems. Numer. Meth. Part. D. E. 24:1301–1320, 2008. doi:10.1002/num.20319 N. Mai-Duy and T. Tran-Cong. Compact local integrated-RBF approximations for second-order elliptic differential problems. J. Comput. Phys. 230:4772–4794, 2011. doi:10.1016/j.jcp.2011.03.002 P. Le, N. Mai-Duy, T. Tran-Cong and G. Baker. A meshless modeling of dynamic strain localization in quasi-brittle materials using radial basis function networks. Comput. Model. Eng. Sci. 25:43–68, 2008. doi:10.3970/cmes.2008.025.043 R. Franke. Scattered data interpolation: Test of some methods. Math. Comput. 38:181–200, 1982. doi:10.1090/S0025-5718-1982-0637296-4

Détail

Publié le : 2018-01-01
DOI : https://doi.org/10.21914/anziamj.v57i0.10446