Parameter-uniform hybrid difference scheme for solutions and derivatives in singularly perturbed initial value problems

A second-order initial value problem with a small parameter multiplying the first- and second-order derivatives is considered. The precise knowledge about the behavior of the exact solution is analyzed. Based on the behavior of the exact solution, a hybrid finite difference scheme on a Shishkin mesh is proposed, which is a combination of the second-order difference scheme on the fine mesh and the modified midpoint upwind scheme on the coarse mesh. By applying the truncation error estimate techniques and a difference analogue of Gronwall's inequality we prove that the scheme is almost second-order convergent for numerical solutions and scaled numerical derivatives. Numerical experiments support these theoretical results and indicate that the estimates are sharp.