A Robust Numerical Scheme via Grid Equidistribution for Singularly Perturbed Delay Partial Differential Equations Arising in Control Theory

In this paper, a model constituting the temperature control of a continuous casting process is considered. This model gives rise to singularly perturbed delay differential equations. This motivated us to construct a robust numerical scheme to solve singularly perturbed parabolic delay convection–diffusion problems exhibiting regular boundary layers. A uniform mesh is used to discretize the domain in the temporal direction, while a non-uniform mesh is used to discretize the spatial variable, which is created by equidistributing a monitor function. For the temporal derivative, the numerical scheme uses the backward Euler scheme, while the space derivative follows the upwind scheme. The truncation error analysis is performed. It is demonstrated that the approach has an optimal error bound and converges uniformly with first-order accuracy in the discrete supremum norm. The theoretical findings are validated by numerical experiments.