Difference scheme of highest accuracy order for a singularly perturbed reaction-diffusion equation based on the solution decomposition method

A Dirichlet problem is considered for a singularly perturbed ordinary differential reaction-diffusion equation. For this problem, a new approach is developed in order to construct difference schemes whose solutions converge in the maximum norm uniformly with respect to the perturbation parameter $\varepsilon$, $\varepsilon \in (0,1]$ (i.e., $\varepsilon$-uniformly) with order of accuracy significantly greater than the achievable accuracy order for the Richardson method on piecewise-uniform grids. Important in this approach is the use of uniform grids for solving grid subproblems for regular and singular components of the grid solution. Using the asymptotic construction technique, a basic difference scheme of the solution decomposition method is constructed that converges $\varepsilon$-uniformly in the maximum norm at the rate ${\mathcal O} \left(N^{-2} \ln^2 N\right)$, where $N+1$ is the number of nodes in the uniform grids used. The Richardson extrapolation technique on three embedded grids is applied to the basic scheme of the solution decomposition method. As a result, we have constructed the Richardson scheme of the solution decomposition method with highest accuracy order. The solution of this scheme converges $\varepsilon$-uniformly in the maximum norm at the rate ${\mathcal O} \left(N^{-6} \ln^6 N\right)$.