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Title: | Numerical procedures in defining entropy solutions for conservation laws Numeričke procedure u definisanju pravilnih rešenja zakona održanja |
Authors: | Krunić Tanja | Keywords: | conservation laws, discontiuous flux function, regularization, discrete shock profiles, singular shock waves;zakoni održanja, prekidna funkcija fluksa, regularizacija, diskretni udarni profili, singularni udarni talasi | Issue Date: | 1-Sep-2016 | Publisher: | Univerzitet u Novom Sadu, Prirodno-matematički fakultet u Novom Sadu University of Novi Sad, Faculty of Sciences at Novi Sad |
Abstract: | <p> U okviru ove doktorske disertacije posmatrani su zakoni održanja sa funkcijom fluksa koja ima prekid u x = 0, pri čemu delovi fluksa levo i desno od x = 0 imaju smo po jedan ekstrem. U prvoj glavi se može naći pregled osnovnih pojmova, definicija i teorema. U drugoj glavi su opisani hiperbolični sistemi zakona održanja, slaba rešenja, kao <br />i numerički postupci za njihovo rešavanje. U trećoj glavi su predstavljeni diskretni profili darnih talasa. U četvrtoj glavi su opisani zakoni održanja sa prekidnom funkcijom fluksa i ukratko su predstvaljeni rezultati drugih autora iz ove oblasti. U petoj glavi je najpre analizirana tzv. jednačina sa dva fluksa u slučaju kada oba dela fluksa levo i desno od x = 0 imaju minimum, a pri tome se seku u najviše jednoj tačci unutar intervala. Primenom regularizacije na intervalu [−<em>ε, ε</em>], za<em> ε</em> > 0 dovoljno malo, dokazano je postojanje diskretnih udarnih profila za postupak Godunova za zakone održanja sa promenljivom funkcijom fluksa. Definisan je i odgovarajući diskretan uslov entropije, a postojanje entropijskog diskretnog udarnog profila je postavljen kao kriterijum za dopustivost udarnih talasa. Potom je analizirana ista jednačina u slucaju kada deo fluksa levo od x = 0 ima maksimum, a deo fluksa desno od x = 0 minimum, dok se oba dela fluksa seku na krajevima posmatranog intervala. U ovom slučaju, uopšten je uslov entropije. U okviru ove glave je prikazano nekoliko numeričkih primera za oba opisana slučaja. Numerički rezultati su dobijeni korišcenjem softvera razvijenog za potrebe ove teze u pro<br />gramskom paketu <em>Mathematica</em>.</p> <p>We consider conservation laws with a flux discontinuity at x = 0, where the flux parts from both left and right hand side of x = 0 have at most one extreme on the observed domain. The first chapter provides elementary definitions and theorems..The second chapter refers to hyperbolic systems of conservation laws, their solutions, and numerical procedures. The third chapter is devoted to discrete shock profiles. The fourth chapter describes conservation laws with discontinuous flux functions and provides basic information upon known results in this field. In the fifth chapter, we first analyse the two-flux equation when both flux parts have a minimum and cross at most at one point in the interior of the domain. Using a flux regularization on the interval [−ε, ε], for ε > 0 small enough, we show the existence of discrete shock profiles for Godunov’s scheme for conservation laws with discontinuous flux functions. We also define a discrete entropy condition accordingly, and use the existence of an entropy discrete shock profile as an entropy criterion for shocks. Then we analyse the same problem in the case when the flux part on the left of x = 0 has a maximum and the part on the right of x = 0 has a minimum, whereas the fluxes cross at the edges of the interval. We derive a more general discrete entropy condition in this case. We provide several numerical examples in both of the above mentioned flux cases. All the presented numerical results are obtained using a program written in Mathematica. Finally, in chapter six, we prove the existence of singular shock waves in the case when the graph of one of the flux parts is above the graph of the other one on the entire domain. For that purpose, we use the shadow wave technique. At the end of this chapter, we provide a numerical verification of the obtained singular solution.</p> |
URI: | https://open.uns.ac.rs/handle/123456789/16828 |
Appears in Collections: | PMF Teze/Theses |
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