Numerical Approximation of Dynamic Models for Special Relativistic Flows

Abstract

This thesis is mainly focused on numerical approximations of the special relativistic hydrodynamic flow models using the central upwind scheme. In recent years, relativistic hydrodynamic models have played a pivotal role in many fields such as nuclear collisions of high energy, astrophysics, laser technology, etc. The precise modeling of numerous features of the high energy in astrophysical phenomenon involves the evaluation of the Einstein theory simultaneously with those of special relativistic hydrodynamic (SRHD) equations. Such numerical models seems to be more complex rather than the nonrelativistic because of nonlinearity relation between the conserved and the primitive quantities. The main goal of the thesis project is to establish a very simple, reliable and efficient numerical techniques in order to solve the special relativistic hydrodynamic (SRHD) and the ultra-relativistic hydrodynamic (URHD) equations. Both single and multi-component flows are considered. A high-resolution shock-capturing central upwind schemes are extended and then applied for solving the governing sets of equations. The proposed numerical algorithm utilizes the accurate information of locally propagating speeds to minimize numerical diffusions in the solutions. This scheme provides the second-order accuracy by applying the MUSCL type reconstruction as well as Runge-Kutta (RK) time step method. After discussing the model equations and solution technique employed, a series of one and two-dimensional numerical test problems are conducted. In order to validate the capability of suggested method and its accuracy, the staggered central (NT) and the kinetic flux vector splitting (KFVS) schemes are also implemented on the equations of same model. Where it is observed that the suggested scheme is robust with less error accuracy as compared to those schemes available in the literature having the sooth algorithms, even in the case of highly-relativistic x two-space dimensional numerical test problems. The major part of this thesis is organized by the following approach. The first chapter of this thesis provided the historical background that inspire me to work on this project. The relativistic hydrodynamics (RHD) simulations have a key role in the astrophysical system to understand the actual mechanism. The importance and uses of these hydrodynamics models which lead towards efficiency and accuracy of various schemes e.g., special relativistic hydrodynamics (SRHD) models are presented in this chapter. The schemes such as the staggered central and central upwind are proposed to numerically execute the special relativistic hydrodynamic model equations. Anticipated numerical achievements of proposed schemes are pointed out in this portion of thesis. Here, the portion gives a brief overview of the state and the conservative formulation techniques of flow models, weak solutions, hyperbolic systems of conservation laws, Riemann problem and nonrelativistic sets of Euler equations. A brief description of relativistic hydrodynamic models in the historical background is also provided in chapter two. The backgrounds of special relativistic hydrodynamic(SRHD) flow models are presented in chapter three. The one and two-dimensional central upwind methods are successfully implemented to estimate the flow model equations. Many test problems are provided by this chapter. The numerical solutions of the central upwind scheme are analyzed with the exact solutions as well as with the solutions obtained from central and kinetic flux vectors splitting schemes in details. Graphical results and the error analysis are also presented. The major findings of chapter three have been published in International journal of PLOS ONE, 10 (2015): e0128698. Chapter four is deal with the approximate solutions of the ultrarelativistic hydrodynamic xi (URHD) models for single and two-space dimension. The scheme of central upwind is proposed in order to approximates the modeled equations. For validation, complicated numerical test cases are carried out. The numerical solutions as well as graphical presentation of the central upwind schemes are analyzed with the available limited results of KFVS and central schemes. The main conclusions of chapter four have been published in Results in Physics, 9 (2018) 1161-1169. In chapter five, we extends the Euler equations for special relativistic flow to multicomponent flow. The central (NT) schemes are implemented to solve for single and two space dimensions models of relativistic multi-component flows. Several numerical problems of RHD model are presented to illustrate the higher accuracy, reliability and efficient of designed schemes are considered here. In order to under stand the results various graphical figures are depicted in this chapter. Furthermore, the key points of the current chapter have been published in Applied Mathematics, 5 (2014) 1169-1186. Chapter six finalize the thesis by summarizing the obtained results and gives outlook to the future work. Chapter seven contains the cited references that have published in various authentic international Journals.