Numerical procedures in defining entropy solutions for conservation laws

Abstract

<p>&nbsp;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&nbsp; glavi su opisani hiperbolični sistemi zakona održanja, slaba re&scaron;enja, kao&nbsp;<br />i numerički postupci za njihovo re&scaron;avanje. U trećoj glavi su predstavljeni&nbsp; diskretni profili darnih talasa. U četvrtoj glavi su opisani zakoni održanja&nbsp; 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&nbsp; x = 0 imaju minimum, a pri tome se seku u najvi&scaron;e jednoj tačci unutar intervala. Primenom regularizacije na intervalu [&minus;<em>&epsilon;, &epsilon;</em>], za<em> &epsilon;</em> &gt; 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&nbsp; udarnog profila je postavljen kao kriterijum za dopustivost udarnih talasa. Potom je analizirana ista jednačina u slucaju kada deo fluksa levo&nbsp; od x = 0 ima maksimum, a deo fluksa desno od x = 0 minimum, dok se oba dela fluksa seku na&nbsp; krajevima posmatranog intervala. U ovom slučaju, uop&scaron;ten je uslov entropije. U okviru ove glave je prikazano nekoliko numeričkih primera za oba opisana slučaja. Numerički rezultati&nbsp; su dobijeni kori&scaron;cenjem 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&nbsp; observed &nbsp;domain. The first chapter provides elementary definitions and theorems..The second chapter refers to hyperbolic systems of conservation laws, their solutions, and&nbsp; numerical procedures. The third chapter is devoted to discrete&nbsp; shock profiles. The fourth chapter describes conservation laws with discontinuous flux functions and provides basic information upon known results in this field. In the&nbsp; fifth chapter, we first &nbsp;analyse the two-flux equation when both flux parts have a minimum and cross at most&nbsp;&nbsp;&nbsp; at one point in the interior of the domain. Using a flux regularization on the interval [&minus;&epsilon;,&nbsp;&nbsp; &epsilon;], for &epsilon; &gt; 0 small enough, we show the existence of discrete shock profiles for Godunov&rsquo;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&nbsp; presented numerical results are obtained using a program written in Mathematica. Finally, in chapter six, we prove the existence of&nbsp; 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>