Scalar Conservation Laws

2019-04-01
We present a theoretical aspect of conservation laws by using simplest scalar models with essential properties. We start by rewriting the general scalar conserva- tion law as a quasilinear partial differential equation and solve it by method of characteristics. Here we come across with the notion of strong and weak solutions depending on the initial value of the problem. Taking into account a special initial data for the left and right side of a discontinuity point, we get the related Riemann problem. An illustration of this problem is provided by some examples. In the remaining part of the chapter, we extend this analysis to the gas dynamics given in the Euler system of equations in one dimension. The transformations of this system into the Lagrangian coordinates follow by applying a suitable change of coordinates which is one of the main issues of this section. We next introduce a first-order Godunov finite volume scheme for scalar conservation laws which leads us to write Godunov schemes in both Eulerian and Lagrangian coordinates in one dimension where, in particular, the Lagrangian scheme is reformulated as a finite volume method. Finally, we end up the chapter by providing a comparison of Eulerian and Lagrangian approaches.

Suggestions

 The Effects of type and amount of organically modified montmorillonites on thermal degradation behavior and mechanical properties of poly(lactic acid) Özdemir, Esra; Hacaloğlu, Jale; Tinçer, Teoman; Department of Polymer Science and Technology (2014) Biodegradable polymers have gained considerable interest as a consequence of significant environmental friendly properties like biocompatibility and ability to synthesis from renewable resources. Even though many biodegradable polymers present comparable properties with petrochemical polymers, there are still many technical barriers like poor thermal resistance, and/or mechanical properties. To overcome these drawbacks, generally, nano-sized materials are incorporated into the continuous matrix. Therefore, ...
 Suya doygun olmayan zeminlerde oturma ve kayma dayanımı Huvaj Sarıhan, Nejan(2014-12-31) Many geotechnical projects involve unsaturated soils. Some of the examples of these are calculation of foundation settlement, bearing capacity on unsaturated clayey soils, slope stability of earth slopes and embankments, excavations. These and many other problems require good understanding of compression and shear behavior of unsaturated soils. However, unsaturated soil testing in the laboratory is not very straightforward because many labs do not have sophisticated equipment and knowledgeable personnel to ...
 An Integrated Approach to Conservation Based on the Interrelations of Tangible and Intangible Cultural Properties Karakul, Özlem (Middle East Technical University, Faculty of Architecture, 2011) Studies for the conservation of historic environments have evolved from the conservation of only physical properties to an inclusive conservation approach concerning cultural properties. The significance of the cultural aspects of historic environments has been realized and discussed especially since 1950-1960s. Despite of an increasing awareness of the subject, the studies on the identification and the documentation of intangible cultural properties are still less advanced than those on tangible cult...
 Probabilistic modeling of operational dynamic water balance system for a mining facility Pelenk, Ümit Giray; Yılmaz, Koray K.; Department of Geological Engineering (2019) Water management in any environment is quite difficult and complex not only because of the effort that is required to comply with the environmental regulations but also due to the dynamic nature of the system not well suited for the deterministic approaches. As a result, probabilistic approaches are developed to make decisions that can actually represent the uncertainties quantitatively as probability distributions. Such methods, considered to be the best practice in mining industry, are very useful to make...
 An Assessment on conservation activities in Bursa, focusing on conservation council decisions 1955 - 2012 Çakıcı, Sermin; Şahin Güçhan, Neriman; Restoration in Department of Architecture (2015) Theoretical, legal and organizational processes in conservation have been widely studied by scholars; however, conservation activities -such as listing procedures and applied conservation plan together with restoration projects in Turkey are poorly documented and published. There are not enough inventories of any kind and literature, such as periodicals on conservation and restoration work, remain mostly inadequate in defining the practical processes of architectural and urban conservation. This lack of inf...
Citation Formats
B. Okutmuştur, Scalar Conservation Laws. 2019, p. 92.