الفهرس 
Models in MathematicsOnly 14 pages are availabe for public view
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Abstract
Hyperbolic conservation laws play a crucial role in elucidating a broad spectrum of phenomena across diverse scientific domains, including compressible gasdynamics, shallow water flow, weather prediction, plasma modeling, rarefied gasdynamics, and beyond. Given the limited availability of analytical solutions forhyperbolic conservation laws, numerical methods have become indispensable forcomprehending these intricate equations. Even in cases where the initial dataexhibits smoothness, solutions to hyperbolic conservation laws may develop discontinuities. Effective numerical methods must accurately locate these discontinuitieswithout introducing unwarranted oscillations and must attain a high level of precision in regions characterized by smooth behavior.While numerical methods of second-order accuracy or higher often exhibit spuriousoscillations near discontinuities, monotone schemes, while preventing oscillations,are constrained to first-order accuracy. Consequently, there has been a concertedeffort to develop high-order numerical schemes to address these challenges.
The main objective of this thesis is improving very high order Weighted EssentiallyNon-Oscillatory (WENO) methods for solving hyperbolic conservation laws, maintaining the non-oscillatory property of ENO near discontinuities, and exhibitingsuperior performance in smooth regions, achieving an accuracy order of (2r-1) insmooth regions when r stencils are employed. Noteworthy advantages of WENOschemes include uniform high-order accuracy in smooth regions, non-oscillatory
shock transitions, robustness for systems with strong shocks, and suitability forsimulating solutions characterized by both discontinuities and intricate smoothstructures, such as shock interactions with vortices. This thesis is structured intofour chapters, supplemented by a bibliography and an Arabic summary. Theorganizational framework of the thesis is as follows:
Chapter 1: This chapter provides a Comprehensive overview of key conceptsrelated to mathematical models governing physical phenomena. It offers a detailedexposition of the problem at hand, explores diverse linear and non-linear finite difference schemes, and conducts an examination of various flux descriptions designedfor high-resolution schemes.
Chapter 2 : This chapter offers a review of the classical fifth-order WENOscheme and how it developed to be more accurate in the perturbational schemeand presents a brief of the classical seventh-order WENO scheme and studies thenew seventh-order perturbational WENO scheme, the theoretical analysis of it,time discretization, and the numerical results.
Chapter 3 : This chapter introduces a brief of the sixth-order WENO schemewith modified stencils and offers the new eighth-order perturbational WENOscheme that produces the atheoretical analysis of it, time discretization, and thenumerical results.
Chapter 4 : In this chapter, we propose the conclusion of the thesis and therecommendation work for other researchers.