Gray-Scott Reaction-Diffusion Cell with an Applied Electric Field

Consider the Gray–Scott reaction-diffusion cell with an applied electric field. The governing equations and boundary and initial conditions are:

,

,

at ,

and at and .

The time series, and at are plotted in orange and blue, respectively. You can vary the values of parameters , , , and . The phase plane diagram is also plotted in magenta.

Various dynamic behaviors are observed: these include (1) a limit cycle and sustained oscillations of and versus time for and (2) a stable node and damped oscillations of and versus time for .

In the discrete Chebyshev–Gauss–Lobatto case, the interior points are given by . These points are the extrema of the Chebyshev polynomials of the first kind, .

The Chebyshev derivative matrix at the quadrature points is an matrix given by

, , for , and for and ,

where for and .

The matrix is then used as follows: and , where is a vector formed by evaluating at , , and and are the approximations of and at the .

Reference

[1] P. Moin, Fundamentals of Engineering Numerical Analysis, Cambridge, UK: Cambridge University Press, 2001.

[2] L. N. Trefethen, Spectral Methods in Matlab, Philadelphia: SIAM, 2000.

[3] A. W. Thornton and T. R. Marchant, "Semi-analytical solutions for a Gray–Scott reaction–diffusion cell with an applied electric field," Chemical Engineering Science, 63(2), 2008 pp. 495–502. DOI: 10.1016/j.ces.2007.10.001 .