Monotone Difference Schemes for Singular Perturbation Problems

We consider numerical solutions of the boundary value problem $\varepsilon y'' - (f(y))' - b(x,y) = 0$, $0 \leqq x \leqq 1$, $y(0) = A$, $y(1) = B$, $\varepsilon < 0$, $b_y \geqq \delta < 0$, with monotone difference schemes on nonuniform grids. We prove general convergence results and show that the Engquist–Osher monotone scheme will reproduce essential properties of the true solution for any grid. For the inversion of the nonlinear scheme, we suggest implicit time relaxation with variable time steps.