Shishkin mesh based septic Hermite interpolation algorithm for time-dependent singularly perturbed convection–diffusion models.

The septic Hermite collocation method is used to examine the behavior of time-dependent singularly perturbed convection–diffusion equations in this paper. The solution of such type of equation contains a sharp boundary layer, i.e., the solution changes rapidly for a small region of the domain when the perturbation parameter (ε ) approaches to zero. The scheme over Shishkin mesh is applied for spatial discretization and for time discretization forward differences are used. The ε -uniform convergence of the algorithm is studied and is found up to sixth order of logarithmic factor in space. The method is shown to be parameter-uniform. The method is implemented on singularly perturbed equations for Dirichlet and Robin boundary conditions. The results are given in the form of 2D, 3D graphs, and tabular forms, which indicates that the proposed method is capable of accurately capturing the sharp boundary layers when perturbation parameter tends to zero. The results exhibit that the proposed technique is efficient, reliable, and works even for a very small values of the perturbation parameter. The numerical results corroborate the theoretical results. [ABSTRACT FROM AUTHOR]