The convergence of a difference scheme approximating a boundary value problem of the hyperbolic type

The difficulty in applying differential problems in practice lies mainly in the impossibility of obtaining their solutions in an analytical form, which makes the development of numerical methods relevant. In this paper, we constructed one implicit difference scheme approximating a boundary value problem of hyperbolic type with homogeneous boundary conditions. The order of approximation of the difference scheme is found. Particular attention is paid to the proof of its stability and convergence. In the proof, an approach similar to the method of separation of variables in mathematical physics was used. The author found the convergence condition imposed on the parameters of the difference scheme. A numerical experiment is carried out. A program has been developed to find and visualize an approximate solution.