An Adaptive Grid Method for Singularly Perturbed Time-Dependent Convection-Diffusion Problems (Communications in Computational Physics, 2020)
In this paper, we study the numerical solution of singularly perturbed time-dependentconvection-diffusion problems. To solve these problems, the backward Eulermethod is first applied to discretize the time derivative on a uniform mesh, and theclassical upwind finite difference scheme is used to approximate the spatial derivativeon an arbitrary nonuniform grid. Then, in order to obtain an adaptive grid for alltemporal levels, we construct a positive monitor function, which is similar to the arc-lengthmonitor function. Furthermore, the ε-uniform convergence of the fully discretescheme is derived for the numerical solution. Finally, some numerical results are givento support our theoretical results.