The Study Of Numerical Algorithms For The Inviscid Burgers Equations With Delay

The inviscid Burgers equations with delay can be used as a variety of mathematics models of physical phenomena, such as atmospheric, traffic dynamics, turbulence. The description of the systems are much closer to the actual situations when we add delay item in the equations. Therefore, the theoretical studies for the equations have much academic value and practical applications. In addition, the exact solutions of generalized nonlinear equations are difficult to solve, and the major way is to get their numerical solutions of the corresponding equations. This paper is devoted to solving a class of inviscid Burgers equations with delay and proposes two types of efficient numerical solvers.In the first chapter, we elaborate the research status of finite difference methods for solving Burgers equations and the delay Burgers equations respectively, as well as the major work in this paper. In the second chapter, we propose a Crank-Nicolson finite difference scheme for the inviscid Burgers equation with delay, which is a two-level implicit scheme. We demonstrate the existence and uniqueness, convergence, stability and boundedness of this difference scheme. Meanwhile, some numerical tests verify our theoretical analysis. In the third chapter, we gain a three-level scheme for the inviscid Burgers equation with delay, where the delay item is disposed by the linear mean interpolation. We also prove a serious of theoretical results, and some numerical tests present the effectiveness of the scheme.In the fourth chapter, we summarize the previous two kinds of numerical methods for solving the inviscid Burgers equations with delay, and find that the three-level difference scheme is superior to the C-N difference scheme when the right-side function f is linear on the unknown term txu),(. In addition, the CPU time of the three-level difference scheme is smaller than the C-N difference scheme obviously. In the end, both of the numerical methods are efficient for dealing with these equations from the results of numerical tests.