Compact-Difference Scheme for the Equation of String Oscillation

The first initial-boundary value problem for the equation of string oscillation with variable coefficients is investigated. For the numerical solution of the problem, a compact-difference scheme of the fourth order of accuracy in h and of the second order of accuracy in τ is constructed. Compact-difference schemes are a powerful tool for numerical modeling of the equation of string oscillation. They allow achieving high accuracy with a relatively small number of computational nodes, which is especially important for problems related to wave processes. An a priori estimate of the solution in the difference interpretation is obtained using the method of energy inequalities. This estimate implies the uniqueness and stability of the solution with respect to the right-hand side and the initial data. Assuming the existence of an exact solution to the original differential problem in the class of sufficiently smooth functions, and also due to the linearity of the problem under consideration, the obtained a priori estimate implies the convergence of the solution of the constructed difference problem to the solution of the original differential problem with a rate of O(h4+τ2) .