This work is devoted to the numerical solution of the acoustic problem. It has a lot of important applications, for example, direct and inverse problems of the seismic survey process. The grid-characteristic approach was used to transform the problem into a set of independent linear transport equations. In one-dimensional case explicit numerical schemes for continuous and discontinuous materials were derived. Using differential sequences of initial equations, the new compact scheme on the two-point spatial stencil was proposed. A set of numerical experiments with different initial conditions and materials were carried out. Direct numerical simulations proved the third order of the approximation.

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