Numerical Studies Of A Shock-wave Problem :

Mostafa Abd Elhamid Ahmed Mahmoud. 

PREFACE AND SUMMERYOne ot the most important topics in comput,tionalFluid Dynamics is the study ot shock and rarefa~tionwaves. In most practical applications, the rel~vant.equations are coupled and non_linear and numeri’cal methodsmust be used to solve these equations. Moreover,additional techniques ( Artiticial Viscosity)ateneededto get rid ot the non-linear oscillations that takeplace in treating shock waves numerically.Tbe aim of this thesis is to study such techniquesand lntroduC.eit in the numerical solutions without addingit explicitly.In cha.pter I, }l’ediscuss physical and mathematicalpictures ot shock waves and the Rankine-Hugoniont relations.The Eulerian, Lagrangian approaches and discretizationin computational Fluid Dynamics are also discussed.In chapter II, we express the quasi-linear hyperbolicsystem, that describe the unsteady compressible floY ofa polytropic. gas, in a non_dimensional Eulerian form. Wealso review some otthe eXplicit, first, second, thirdand tourth order accurate method, tha.t. ·are frequently .used to solve the system numerically. The stability ofsome ot these methods. using both Hirt and von_Neumannapproaches ifla~so discussed. In,the ~ast-part ot chapterII. ye revieY tbe ditterent artiticia~ viscosity tormsthat, 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. asa rep~acement to the artitica~ viscosity. ~he samescheme vas used by E~_Fayoumi and Fkirin (~978) to iteratetirst order numerica~ so~utions otthe quasi-~inear systemthat represent the compressib~e one dimensiona~ t~owot a pO~ytropic gas in cartizian coordinates.Here. ye iterate second order Lax~wendrott so~utionof 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 titthinternational”~ongress ~or statistics. computer science.socia~ ,and demographiC res~arch” (pp. 433-448)”~rganizedby Ain ShamS University. 29 March-3 Apri~ ~980.~he same article has been accep”ted”to~ Jl1Jb:l:tc”t”tonin the EgyptlanComputer Journa~.{ ~o appear in June ~980 Ii-lfiu.1e.