الفهرس 
IntroductionOnly 14 pages are availabe for public view
 |  | from | 147 |  |  |
| | | from | 147 | | |
Abstract
Many areas of practical interests e.g., aero-dynamics gas-dynamics, fluid dynamics, can be modelled by the time dependent partial differential equations of hyperbolic type known as hyperbolic conservation laws.
It is well known that the analytical solutions of these problems are available only in a few model problems and one thus has to rely on numerical methods for solving problems of practical interest. The solutions of hyperbolic equations laws may develop discontinuities even when the initial condition is smooth. Therefore, the numerical method should compute such discontinuities with the correct position and without spurious oscillations and yet achieve high order of accuracy in the smooth regions.
Though the solution of linear hyperbolic problems does not have discontinuities if the initial data is smooth and one may expect the numerical treatment of such linear problems to be easy, but in practice this is not the case. Many of the well-known schemes are reported to fail even for linear problems. In case of non-linear problems, situation becomes worse and more complex problems like non-linear instability and convergence to wrong weak solution known as entropy violating solutions arise. Though many researchers contributed to solve hyperbolic problems and gave elegant theories and numerical algorithms but still this area stimulates a great deal of research and attracts researchers of various engineering streams and Mathematics.
The present work is divided into five chapters.
Chapter 1: We give a brief review of some mathematical models and survey of the hyperbolic equations along with the numerical methods used to solve the problem.
Chapter 2: In this chapter, a new third order finite difference scheme, for solving initial value problems for conservation laws, is introduced.
Chapter 3: We propose in this chapter a new eighth order adaptive central-upwind WENO scheme. This scheme, recovers the eighth order central scheme with optimal weights and adapts between the central and upwind schemes smoothly by a new weighting strategy based on the smoothness indicators of the optimal higher order stencil and lower order upwind stencils
Chapter 4, we present the new central seventh order WENO reconstruction the main advantages of this method are the choice of the ideal weights has no effect on the accuracy and simplicity for computing the ideal weights.
Chapter 5: In this chapter we present the knew hybrid scheme uses a more general central-upwind scheme, universal and efficient scheme. For the time integration we use the third order TVD Runge-Kutta method.