Layer resolving fitted mesh method for parabolic convection-diffusion problems with a variable diffusion

In this paper, we constructed a fitted mesh finite difference method for solving a class of time-dependent singularly perturbed turning point convection-diffusion problems whose solution exhibits an interior layer. The diffusion coefficient in the underlying PDE is a quadratic function of the space variable and contains a perturbation parameter. While such problems have been studied in the case of boundary layers, little has been achieved for interior layer problems where the coefficient functions are considered to be dependent on the space variable alone. In this work, we focus our attention to such problems where the coefficient functions are dependent of both the space and time variables. Following the work of Liseikin (USSR Computational Mathematics and Mathematical Physics 26(6), 133–139, 1986), we establish bounds on the solution and its derivatives. Then we discretize the time derivative using an implicit Euler method. This discretization results in a set of two-point boundary value problems (TPBVPs). We then construct a fitted mesh finite difference method to solve these TPBVPs. This method is analyzed for stability and convergence. We proved that it satisfies a minimum principle and is uniformly convergent with respect to the perturbation parameter. In order to improve the accuracy of the proposed method, we use the Richardson extrapolation. Finally, we present some numerical experiments to validate our theoretical findings.