The main disadvantage of explicit schemes for the numerical solution to nonstationary problems is in a very strong stability condition for the size of a time step size. One of the possibilities to improve the efficiency of explicit algorithms is to use different time steps in different space subdomains. From this point of view the methods studied below can be considered as a special case of domain decomposition methods. This approach allows increasing the accuracy of results for the problems corresponding to multiscale physical processes. A striking example is the problem of the laminar flame propagation. There are two natural subdomains in this problem: the subdomain corresponding to the area of diffusion processes and the subdomain corresponding to the kinetic area. The latter is quite narrow and requires a very small spatial step to attain an admissible accuracy. The schemes with the time steps variable in space are studied in [1—3] for the implicit schemes. In [4, 5], a similar technique was applied to provide the localization of a stability condition in subdomains for the explicit schemes. The Dirichlet and the Neumann boundary conditions were used at the interface of the subdomains in [4] and [5], respectively. Applications of these methods are presented in papers [6, 7], and in [8], aspects of parallelization are discussed.

In this paper, we have improved the results from [4]. Namely, we demonstrate the estimate of stability with respect to the right hand-side independent of the number of interior layers (see below).