Layer-adapted methods for quasilinear singularly perturbed delay differential problems

In this work we consider a class of initial value problems for quasilinear singularly perturbed first order delay differential equations. To solve this class of problems numerically we consider two finite difference schemes: the backward Euler scheme and a high order hybrid scheme which is a blend of the Trapezoidal scheme and the backward Euler scheme. We derive general convergence results for both the schemes, based on which a number of layer-adapted meshes can be constructed and analyzed. As consequences of these results we establish uniform convergence of the schemes on certain layer-adapted meshes. Numerical experiments confirm our theoretical findings.