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Abstract PREFACE AND SUMMERY One ot the most important topics in comput,tional Fluid Dynamics is the study ot shock and rarefa~tion waves. In most practical applications, the rel~vant .equations are coupled and non_linear and numeri’cal methods must be used to solve these equations. Moreover, additional techniques ( Artiticial Viscosity)at e needed to get rid ot the non-linear oscillations that take place in treating shock waves numerically. Tbe aim of this thesis is to study such techniques and lntroduC.eit in the numerical solutions without adding it explicitly. In cha.pter I, }l’ediscuss physical and mathematical pictures ot shock waves and the Rankine-Hugoniont relations. The Eulerian, Lagrangian approaches and discretization in computational Fluid Dynamics are also discussed. In chapter II, we express the quasi-linear hyperbolic system, that describe the unsteady compressible floY of a polytropic. gas, in a non_dimensional Eulerian form. We also review some otthe eXplicit, first, second, third and tourth order accurate method, tha.t. ·are frequently . used to solve the system numerically. The stability of some ot these methods. using both Hirt and von_Neumann approaches ifla~so discussed. In,the ~ast-part ot chapter II. ye revieY tbe ditterent artiticia~ viscosity torms that, are used to e~iminate PQst shock osci~~ations. In,chapter III. we moditied an ,iterative sche’lle. origina~~y devised by Abarbane~ and Zyas in ~969. as a rep~acement to the artitica~ viscosity. ~he same scheme vas used by E~_Fayoumi and Fkirin (~978) to iterate tirst order numerica~ so~utions otthe quasi-~inear system that represent the compressib~e one dimensiona~ t~ow ot a pO~ytropic gas in cartizian coordinates. Here. ye iterate second order Lax~wendrott so~ution of the system that represent the compressib~e one di~en- , siona~ t~ow ot a pO~ytropic gas in ct~il;ldrica~co~rdinates’. Chapter III has been pup~ished in the preoceeding ot titth international”~ongress ~or statistics. computer science. socia~ ,and demographiC res~arch” (pp. 433- 448 )”~rganized by Ain ShamS University. 29 March-3 Apri~ ~980. ~he same article has been accep”ted”to~ Jl1Jb:l:t c ”t”to n in the EgyptlanComputer Journa~. { ~o appear in June ~980 Ii-lfiu.1e. |