Improved approximations of the solution and derivatives to a singularly perturbed reaction-diffusion equation based on the solution decomposition method

In the case of the Dirichlet problem for a singularly perturbed ordinary differential reaction–diffusion equation, a new approach is used to the construction of finite difference schemes such that their solutions and their normalized first- and second-order derivatives converge in the maximum norm uniformly with respect to a perturbation parameter $\varepsilon\in(0,1]$; the normalized derivatives are $\varepsilon$-uniformly bounded. The key idea of this approach to the construction of $\varepsilon$-uniformly convergent finite difference schemes is the use of uniform grids for solving grid subproblems for the regular and singular components of the grid solution. Based on the asymptotic construction technique, a scheme of the solution decomposition method is constructed such that its solution and its normalized first- and second-order derivatives converge $\varepsilon$-uniformly at the rate of $O(N^{-2}\ln^2N)$, where $N+1$ is the number of points in the uniform grids. Using the Richardson technique, an improved scheme of the solution decomposition method is constructed such that its solution and its normalized first and second derivatives converge $\varepsilon$-uniformly in the maximum norm at the same rate of $O(N^{-4}\ln^4N)$.