Generalized Riemann Problem on the Decay of a Discontinuity with Additional Conditions at the Boundary and Its Application for Constructing Computational Algorithms

• Published in: Journal of mathematical sciences (New York, N.Y.) Vol. 263; no. 4; pp. 498 - 510
• Main Authors: Skalko, Yu. I., Gridnev, S. Yu
• Format: Journal Article
• Language: English
• Published: New York Springer US 09.05.2022; Springer; Springer Nature B.V
• Subjects: Algorithms; Boundary value problems; Cauchy problems; Decay; Differential equations; Discontinuity; Hyperbolic systems; Mathematics; Mathematics and Statistics; generalized function; 35L40; 35L50; 35L67; 35L45; characteristic; Cauchy problem; decay of a discontinuity; conjugation condition; equation of elastic dynamics; Riemann invariant; Green matrix-function; hyperbolic system
• ISSN: 1072-3374; 1573-8795
• DOI: 10.1007/s10958-022-05945-2

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 decay 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.