Hyperbolic reaction-diffusion equation for a reversible Brusselator: Solution by a spectral method

Stability characteristics of hyperbolic reaction-diffusion equations with a reversible Brusselator model are investigated as an extension of the previous work. Intensive stability analysis is performed for three important parameters, N-rd, beta and D-x, where N-rd is the reaction-diffusion number which is a measure of hyperbolicity, beta is a measure of reversibility of autocatalytic reaction and D-x is a diffusion coefficient of intermediate X. Especially, the dependence on N-rd of stability exhibits some interesting features, such as hyperbolicity in the small N-rd region and parabolicity in the large N-rd region. The hyperbolic reaction-diffusion equations are solved numerically by a spectral method which is modified and adjusted to hyperbolic partial differential equations. The numerical method gives good accuracy and efficiency even in a stiff region in the case of small N-rd, and it can be extended to a two-dimensional system. Four types of solution, spatially homogeneous, spatially oscillatory, spatio-temporally oscillatory and chaotic can be obtained. Entropy productions for reaction are also calculated to get some crucial information related to the bifurcation of the system. At the bifurcation point, entropy production changes discontinuously and it shows that different structures of the system have different modes in the dissipative process required to maintain the structure of the system. But it appears that magnitude of entropy production in each structure give no important information related for states of system itself.