Geodesic
Generalized Riemann problem on the breakup of a discontinuity with additional conditions at the boundary and its application for constructing computational algorithms
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh Winter Mathematical School "Modern Methods of Function Theory and Related Problems." January 28 – February 2, 2019. Part 1, Tome 170 (2019), pp. 38-50.

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We construct an approximation of the fundamental solution of a problem for a hyperbolic system of first-order linear differential equations with constant coefficients. We propose an algorithm for an approximate solution of the generalized Riemann problem on the breakup of a discontinuity under additional conditions at the boundaries, which allows one to reduce the problem of finding the values of variables on both sides of the discontinuity surface of the initial data to the solution of a system of algebraic equations. We construct a computational algorithm for an approximate solution of the initial-boundary-value problem for a hyperbolic system of first-order linear differential equations. The algorithm is implemented for a system of equations of elastic dynamics; it is used for solving some applied problems associated with oil production.
Keywords: breakup of a discontinuity, hyperbolic system, generalized function, Cauchy problem, Green matrix-function, characteristic, equation of elastic dynamics.
Mots-clés : conjugation condition, Riemann invariant 
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Yu. I. Skalko; S. Yu. Gridnev. Generalized Riemann problem on the breakup of a discontinuity with additional conditions at the boundary and its application for constructing computational algorithms. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh Winter Mathematical School "Modern Methods of Function Theory and Related Problems." January 28 – February 2, 2019. Part 1, Tome 170 (2019), pp. 38-50. http://geodesic.mathdoc.fr/item/INTO_2019_170_a3/

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