A finite difference scheme of improved accuracy on a priori adapted grids for a singularly perturbed parabolic convection–diffusion equation

In the case of the Dirichlet problem for a singularly perturbed parabolic convection–diffusion equation with a small parameter $\varepsilon$ multiplying the higher order derivative, a finite difference scheme of improved order of accuracy that converges almost $\varepsilon$-uniformly (that is, the convergence rate of this scheme weakly depends on $\varepsilon$) is constructed. When $\varepsilon$ is not very small, this scheme converges with an order of accuracy close to two. For the construction of the scheme, we use the classical monotone (of the first order of accuracy) approximations of the differential equation on a priori adapted locally uniform grids that are uniform in the domains where the solution is improved. The boundaries of such domains are determined using a majorant of the singular component of the grid solution. The accuracy of the scheme is improved using the Richardson technique based on two embedded grids. The resulting scheme converges at the rate of $O((\varepsilon^{-1}N^{-K}\ln^2N)^2+N^{-2}\ln^4N+N^{-2}_0)$ as $N,N_0\to\infty$, where $N$ and $N_0$ determine the number of points in the meshes in $x$ and in $t$, respectively, and $K$ is a prescribed number of iteration steps used to improve the grid solution. Outside the $\sigma$-neighbourhood of the lateral boundary near which the boundary layer arises, the scheme converges with the second order in $t$ and with the second order up to a logarithmic factor in $x$; here, $\sigma=O(N^{-(K-1)}\ln^2N)$. The almost $\varepsilon$-uniformly convergent finite difference scheme converges with the defect of $\varepsilon$-uniform convergence $\nu$, namely, under the condition $N^{-1}\ll\varepsilon^{\nu}$, where $\nu$ determining the required number of iteration steps $K$ ($K=K(\nu)$) can be chosen sufficiently small in the interval (0, 1]. When $\varepsilon^{-1}=O(N^{K-1})$, the scheme converges at the rate of $O(N^{-2}\ln^4N+N^{-2}_0)$.