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A second-order accurate numerical method has been proposed for the solution of a coupled non-linear oscillator featuring in chemical kinetics. Although implicit by construction, the method enables the solution of the model initial-value problem (IVP) to be computed explicitly. The second-order method is constructed by taking a linear combination of first-order methods. The stability analysis of the system suggests the existence of a Hopf bifurcation, which is confirmed by the numerical method. Both the critical point of the continuous system and the fixed point of the numerical method will be seen to have the same stability properties. The second-order method is more competitive in terms of numerical stability than some well-known standard methods (such as the Runge–Kutta methods of order two and four). [ABSTRACT FROM AUTHOR]

A second-order method is developed for the numerical solution of the initial-value problems $$u' \equiv du/dt = f_1 \left( {u,v} \right)$$ , $$t > 0$$ , $$u\left( 0 \right) = U^0 $$ and $$v' \equiv dv/dt = f_2 \left( {u,v} \right)$$ , $$t > 0$$ , $$v\left( 0 \right) = V^0 $$ , in which the functions $$f_1 \left( {u,v} \right) = B + u^2 v - \left( {A + 1} \right)u$$ and $$f_2 \left( {u,v} \right) = Au - u^2 v$$ , 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 $$1 - A + B^2 \geqslant 0$$ . 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). [ABSTRACT FROM AUTHOR]