Uniformly convergent method for two coupled nonlinear singularly perturbed systems arising in chemical kinetics

The initial value problem for the nonlinear system of singularly perturbed differential equations, which emerges as a model for chemical kinetics, is considered. In order to solve this problem numerically, a novel fitted difference scheme is constructed by the finite difference method on non-uniform meshes, like the Shishkin mesh and the Bakhvalov mesh, using quadrature rules with the remaining terms in integral form. The scheme is proven to achieve almost first-order convergence in the discrete maximum norm on the Shishkin mesh and first-order convergence on the Bakhvalov mesh. Two numerical examples are considered to illustrate the accuracy and performance of the method. In order to show the advantage of the proposed method we compare our results with those obtained by an implicit linear difference scheme. Comparison shows that the proposed method is fast convergent and highly accurate.