A second-order scheme for the "Brusselator" reaction-diffusion system

A second-order method is developed for the numerical solution of the initia l-value problems u'=du/dt=f(1)(u,v), t>0, u(0)=U-0 and v'=dv/dt=f(2)(u,v), t>0, v(0)=V-0, in which the functions f(1)(u,v)=B+u(2)v-(A+1)u and f(2)(u,v )=Au-u(2)v, where A and B are positive real constants, are the reaction ter ms arising from the mathematical modelling of chemical systems such as in e nzymatic reactions and plasma and laser physics in multiple coupling betwee n modes. The method is based on three first-order methods for solving u and v, respectively. In addition to being second-order accurate in space and t ime, the method is seen to converge to the correct fixed point (U*=B, V*=A/ B) provided 1-A+B(2)greater than or equal to 0. The approach adopted is ext ended to solve a class of non-linear reaction-diffusion equations in two-sp ace dimensions known as the "Brusselator" system. The algorithm is implemen ted in parallel using two processors, each solving a linear algebraic syste m as opposed to solving non-linear systems, which is often required when in tegrating non-linear partial differential equations (PDEs).