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

A second-order method is developed for the numerical solution of the initial-value problems ′ ≡ du/dt = f₁(u, v), t > 0, u(0) = U⁰ and v′ ≡ dv/dt = f₂(u, v), t > 0, v(0) = V⁰, in which the functions f₁(u, v) = B + u²v - (A + 1)u and f₂(u, v) = Au - u²v, where A and B are positive real constants, are the reaction terms arising from the mathematical modelling of chemical systems such as in enzymatic reactions and plasma and laser physics in multiple coupling between 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 time, the method is seen to converge to the correct fixed point (U^* = B, V^* = A/B) provided 1 - A + B² ≥ 0. The approach adopted is extended to solve a class of non-linear reaction-diffusion equations in two-space dimensions known as the "Brusselator" system. The algorithm is implemented in parallel using two processors, each solving a linear algebraic system as opposed to solving non-linear systems, which is often required when integrating non-linear partial differential equations (PDEs).