The Shock Wave Solution Of The Riemann Problem For The Burgers Equation With The Linear Forcing Term

In this paper, the Riemann problem for the inviscid Burgers equation with a linearforcing term is considered. The shock wave solution is obtained by combining thegeneralized Rankine-Hugoniot jump condition together with the method ofcharacteristics, which reflects the impact of the inhomogeneous forcing term on theshock front. In addition, some interesting phenomena are also observed during theconstruction process of Riemann solution when convenient forcing terms are considered.Then the regularization of the shock wave solution to the Riemann problem for therelativistic Burgers equation is considered. For Riemann initial data consisting of asingle decreasing jump, it can be found that the regularization of nonlinear convectiveterm cannot capture the correct shock wave solution. In order to overcome it, a newregularization technique called the observable divergence method introduced byMohseni is considered and it can capture the correct shock wave solution. In addition,Helmholtz filter is taken for the fully explicit computation.This thesis can be divided into five chapters.The first chapter introduces the background and methods of this paper. It includesthe related history and the research at home and abroad for the hyperbolic conservationlaws system and Riemann problem and so on. And the main research of this paper isintroduced to make the readers understand it simply.The second chapter is the basic definitions and theorems of the hyperbolicconservation laws system and Riemann problem and so on which are the foundation ofthis thesis.The third chapter is mainly concerned with the shock wave solution to theRiemann problem for the Burgers equation with the linear forcing term. Firstly, thegeneralized Rankine-Hugoniot jump condition is introduced and proved in the sense ofdistributions. Then we can get the shock wave solution to the Riemann problem with thejump condition and the method of characteristics when two special forcing terms arechosen. Finally we describe the influence of the forcing terms to the shock front indetail through the picture.The forth chapter is devoted to the regularization of the shock wave solution to the Riemann problem for the relativistic Burgers equation. Firstly, it can be found that theregularization of nonlinear convective term cannot capture the correct shock wavesolution to the Riemann problem for the relativistic Burgers equation. So a newregularization technique called the observable divergence method is considered and itcan capture the correct shock wave solution when the Helmholtz filter is taken for thefully explicit computation.In the fifth chapter, we make a summary and outlook to the contents of thisdissertation.