A difference scheme of the decomposition method for an initial boundary value problem for the singularly perturbed transport equation

An initial boundary value problem for the singularly perturbed transport equation is considered. A new approach to constructing the difference scheme based on a special decomposition of solution into the sum of a regular and a singular components is proposed. A difference scheme is constructed based on the solution decomposition method in which the regular and singular components of the solution are considered on uniform grids, and their $\varepsilon$-uniform convergence in the maximum norm with the first order of the convergence rate is proved. Given the grid solutions for the components, a continual solution that approximates the solution of the initial boundary value problem for the singularly perturbed transport equation is constructed, and its $\varepsilon$-uniform convergence in the maximum norm with the first order of the convergence rate is proved. The proposed approach will make it possible to use the technique of improving the convergence rate of grid solutions on embedded grids for constructing difference schemes that converge $\varepsilon$-uniformly with the second-order rate and higher for the initial boundary value problem for the singularly perturbed transport equation.