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

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 additio...

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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
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Abstract 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.
AbstractList 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.
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. Keywords and phrases: decay of a discontinuity, conjugation condition, hyperbolic system, generalized function, Cauchy problem, Green matrix-function, characteristic, Riemann invariant, equation of elastic dynamics. AMS Subject Classification: 35L40, 35L67, 35L45, 35L50
Audience Academic
Author Skalko, Yu. I.
Gridnev, S. Yu
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  fullname: Gridnev, S. Yu
  organization: Voronezh State Technical University
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Cites_doi 10.20537/2076-7633-2014-6-3-347-356
10.1017/CBO9780511791253
10.1111/j.1365-246X.2006.03051.x
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Issue 4
Keywords generalized function
35L40
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characteristic
Cauchy problem
decay of a discontinuity
conjugation condition
equation of elastic dynamics
Riemann invariant
Green matrix-function
hyperbolic system
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KaserMDumbserMAn arbitrary high order discontinuous Galerkin method for elastic waves on unstructured meshes. I. The two-dimensional isotropic case with external source termsGeophys. J. Int.200616685587710.1111/j.1365-246X.2006.03051.x
GelfandIMShilovGEGeneralized Functions. I. Properties and Operations1964New York–LondonAcademic Press
KulikovskyAGPogorelovNVSemenovAYMathematical Problems in Numerical Simulation of Hyperbolic Systems2001MoscowFizmatlit[in Russian]
SkalkoYIRiemann problem of a discontinuity decay in the case of several spatial variablesTr. Mosk. Fiz.-Tekh. Inst.201684169182
SkalkoYICorrect conditions on the boundary separating subdomainsComput. Issled. Model.20146334735610.20537/2076-7633-2014-6-3-347-356
VladimirovVSEquations of Mathematical Physics1981MoscowNauka[in Russian]
LeVequeRLFinite Volume Methods for Hyperbolic Problems2002CambridgeCambridge Univ. Press10.1017/CBO9780511791253
GelfandIMShilovGEGeneralized Functions. II. Spaces of Fundamental and Generalized Functions1968New York–LondonAcademic Press
IM Gelfand (5945_CR3) 1968
M Kaser (5945_CR4) 2006; 166
RL LeVeque (5945_CR6) 2002
YI Skalko (5945_CR7) 2016; 8
YI Skalko (5945_CR8) 2014; 6
VS Vladimirov (5945_CR9) 1981
AG Kulikovsky (5945_CR5) 2001
IM Gelfand (5945_CR1) 1958
IM Gelfand (5945_CR2) 1964
References_xml – reference: SkalkoYIRiemann problem of a discontinuity decay in the case of several spatial variablesTr. Mosk. Fiz.-Tekh. Inst.201684169182
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– reference: KaserMDumbserMAn arbitrary high order discontinuous Galerkin method for elastic waves on unstructured meshes. I. The two-dimensional isotropic case with external source termsGeophys. J. Int.200616685587710.1111/j.1365-246X.2006.03051.x
– reference: SkalkoYICorrect conditions on the boundary separating subdomainsComput. Issled. Model.20146334735610.20537/2076-7633-2014-6-3-347-356
– reference: VladimirovVSEquations of Mathematical Physics1981MoscowNauka[in Russian]
– reference: GelfandIMShilovGEGeneralized Functions. II. Spaces of Fundamental and Generalized Functions1968New York–LondonAcademic Press
– reference: LeVequeRLFinite Volume Methods for Hyperbolic Problems2002CambridgeCambridge Univ. Press10.1017/CBO9780511791253
– reference: GelfandIMShilovGESome Problems of the Theory of Differential Equations1958MoscowGIFML[in Russian]
– reference: KulikovskyAGPogorelovNVSemenovAYMathematical Problems in Numerical Simulation of Hyperbolic Systems2001MoscowFizmatlit[in Russian]
– volume-title: Equations of Mathematical Physics
  year: 1981
  ident: 5945_CR9
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  year: 1958
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SubjectTerms Algorithms
Boundary value problems
Cauchy problems
Decay
Differential equations
Discontinuity
Hyperbolic systems
Mathematics
Mathematics and Statistics
Title Generalized Riemann Problem on the Decay of a Discontinuity with Additional Conditions at the Boundary and Its Application for Constructing Computational Algorithms
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