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Numerical solution of singular patterns in one-dimensional Gray-Scott-like models. (English) Zbl 1401.65101

Summary: In this paper, numerical simulations of coupled one-dimensional Gray-Scott model for pulse splitting process, self-replicating patterns and unsteady oscillatory fronts associated with autocatalytic reaction-diffusion equations as well as homoclinic stripe patterns, self-replicating pulse and other chaotic dynamics in Gierer-Meinhardt equations [A. Doelman et al., Methods Appl. Anal. 8, No. 3, 387–414 (2001; Zbl 1006.92008)] are investigated. Our major approach is the use of higher order exponential time differencing Runge-Kutta (ETDRK) scheme that was earlier proposed by S. M. Cox and P. C. Matthews [J. Comput. Phys. 176, No. 2, 430–455 (2002; Zbl 1005.65069)], which was later presented as a result of instability in a modified form by S. Krogstad [J. Comput. Phys. 203, No. 1, 72–88 (2005; Zbl 1063.65097)] to solve stiff semi-linear problems. The semi-linear problems under consideration in this context are split into linear, which harbors the stiffest part of the dynamical system and nonlinear part that varies slowly than the linear part. For the spatial discretization, we employ higher-order symmetric finite difference scheme and solve the resulting system of ODEs with higher-order ETDRK method. Numerical examples are given to illustrate the accuracy and implementation of the methods, results and error comparisons with other standard schemes are well presented.

MSC:

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
92-08 Computational methods for problems pertaining to biology
35K40 Second-order parabolic systems
35K57 Reaction-diffusion equations
76R50 Diffusion
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