A method of asymptotic constructions of improved accuracy for a quasilinear singularly perturbed parabolic convection-diffusion equation

The Dirichlet problem on an interval for quasilinear singularly perturbed parabolic convection-diffusion equation is considered. The higher order derivative of the equation is multiplied by a parameter $\varepsilon$ that takes any values from the half-open interval $(0,1]$. For this type of linear problems, the order of the $\varepsilon$-uniform convergence (with respect to $x$ and $t$) for the well-known schemes is not higher than unity (in the maximum norm). For the boundary value problem under consideration, grid approximations are constructed that converge $\varepsilon$-uniformly at the rate of $O(N^{-2}\ln^2N+N_0^{-2})$, where $N+1$ and $N_0+1$ are the numbers of the mesh points with respect to $x$ and $t$, respectively. On the $x$ axis, piecewise uniform meshes that condense in the boundary layer are used. If the parameter value is small compared to the effective step of the spatial grid, the domain decomposition method is used, which is motivated by “asymptotic constructions”. Monotone approximations of “auxiliary” subproblems describing the main terms of the asymptotic expansion of the solution outside a neighborhood of the boundary layer neighborhood are used. In the neighborhood of the boundary layer (of the width $O(\varepsilon\ln N)$) the first derivative with respect to $x$ is approximated by the central difference derivative. These subproblems are successively solved in the subdomains on uniform grids. If the parameter values are not sufficiently small (compared to the effective step of the mesh with respect to $x$), the classical implicit difference schemes approximating the first derivative with respect to $x$ by the central difference derivative are applied. To improve the accuracy in $t$, the defect correction technique is used. Notice that the calculation of the solution of the constructed difference scheme (the scheme based on the method of asymptotic constructions) can be considerably simplified for sufficiently small values of the parameter $\varepsilon$.