Explosive behavior in spatially discrete reaction-diffusion systems (Q1045753)

The authors study the evolution problem \[ \frac{\partial u}{\partial t}= d\;\frac{\partial^{2} u}{\partial x^{2}}+v\;\frac{\partial u}{\partial x}+f(u), \] that is discretizied by the spatial variable and take a the form of an ordinary differential equation \[ u'_{n}=d_{n}(u_{n+1}-2u_{n}+u_{n-1})+v_{n}(u_{n-1}-u_{n})+f(u_{n}). \] This equation is solved using Runge-Kutta numerical methods of order 4 and 5 for different reactive sources \( f(u)=u^{3},\quad f(u)=e^{u}.\) The explosive behavior of these solutions is shown for different data in the approximate scheme.