Riemann Problems and Exact Solutions for the p-System
In this paper, within the framework of the Method of Differential Constraints, the celebrated p-system is studied. All the possible constraints compatible with the original governing system are classified. In solving the compatibility conditions between the original governing system and the appended...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-03-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/10/6/935 |
| _version_ | 1797445420280971264 |
|---|---|
| author | Natale Manganaro Alessandra Rizzo |
| author_facet | Natale Manganaro Alessandra Rizzo |
| author_sort | Natale Manganaro |
| collection | DOAJ |
| description | In this paper, within the framework of the Method of Differential Constraints, the celebrated p-system is studied. All the possible constraints compatible with the original governing system are classified. In solving the compatibility conditions between the original governing system and the appended differential constraint, several model laws for the pressure <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>(</mo><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula> are characterised. Therefore, the analysis developed in the paper has been carried out in the case of physical interest where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>=</mo><msub><mi>p</mi><mn>0</mn></msub><msup><mi>v</mi><mrow><mo>−</mo><mi>γ</mi></mrow></msup></mrow></semantics></math></inline-formula>, and an exact solution that generalises simple waves is determined. This allows us to study and to solve a class of generalised Riemann problems (GRP). In particular, we proved that the solution of the GRP can be discussed in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula> plane through rarefaction-like curves and shock curves. Finally, we studied a Riemann problem with structure and we proved the existence of a critical time after which a GRP is solved in terms of non-constant states separated by a shock wave and a rarefaction-like wave. |
| first_indexed | 2024-03-09T13:25:34Z |
| format | Article |
| id | doaj.art-b4783a9b3e0f459d87cd26cb8e974ca5 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T13:25:34Z |
| publishDate | 2022-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-b4783a9b3e0f459d87cd26cb8e974ca52023-11-30T21:24:19ZengMDPI AGMathematics2227-73902022-03-0110693510.3390/math10060935Riemann Problems and Exact Solutions for the p-SystemNatale Manganaro0Alessandra Rizzo1Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, Viale F. Stagno d’Alcontres 31, 98166 Messina, ItalyDepartment of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, Viale F. Stagno d’Alcontres 31, 98166 Messina, ItalyIn this paper, within the framework of the Method of Differential Constraints, the celebrated p-system is studied. All the possible constraints compatible with the original governing system are classified. In solving the compatibility conditions between the original governing system and the appended differential constraint, several model laws for the pressure <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>(</mo><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula> are characterised. Therefore, the analysis developed in the paper has been carried out in the case of physical interest where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>=</mo><msub><mi>p</mi><mn>0</mn></msub><msup><mi>v</mi><mrow><mo>−</mo><mi>γ</mi></mrow></msup></mrow></semantics></math></inline-formula>, and an exact solution that generalises simple waves is determined. This allows us to study and to solve a class of generalised Riemann problems (GRP). In particular, we proved that the solution of the GRP can be discussed in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula> plane through rarefaction-like curves and shock curves. Finally, we studied a Riemann problem with structure and we proved the existence of a critical time after which a GRP is solved in terms of non-constant states separated by a shock wave and a rarefaction-like wave.https://www.mdpi.com/2227-7390/10/6/935Riemann problemsp-systemexact solutions |
| spellingShingle | Natale Manganaro Alessandra Rizzo Riemann Problems and Exact Solutions for the p-System Mathematics Riemann problems p-system exact solutions |
| title | Riemann Problems and Exact Solutions for the p-System |
| title_full | Riemann Problems and Exact Solutions for the p-System |
| title_fullStr | Riemann Problems and Exact Solutions for the p-System |
| title_full_unstemmed | Riemann Problems and Exact Solutions for the p-System |
| title_short | Riemann Problems and Exact Solutions for the p-System |
| title_sort | riemann problems and exact solutions for the p system |
| topic | Riemann problems p-system exact solutions |
| url | https://www.mdpi.com/2227-7390/10/6/935 |
| work_keys_str_mv | AT natalemanganaro riemannproblemsandexactsolutionsforthepsystem AT alessandrarizzo riemannproblemsandexactsolutionsforthepsystem |