Parallel Upstream Weighted Alternating Explicit-Implicit Method For Burgers Equation

Burgers equation is a simple mathematic model for the hydrokinetics Navier-Stokes equation, and it can be considered as the mathematic model for the tra?c ?owdynamics. So it has the academic value for its extensive background in application.Many papers have solved the Burgers equation by ?nite di?erence method, ?niteelement method, boundary ?nite element etc. But explicit scheme is limited bythe strict stability condition, and implicit scheme, though out of that limitation ofstability condition, requires solving large-scale sparse nonlinear equation systems. Sothese methods are un?t for using on parallel or vector computers.In this paper, we mainly discuss a numerical method with intrinsic parallelismfor Burgers equation upstream weighted alternating segment(block) explicit im-plicit method.The method is that,on each time level, the area is divided into a numberof segments(blocks),and on each segment(block), di?erence equation systems is de-signed to be solved independently, so it has intrinsic parallelism, and can be used onparallel computer of high-powered multiprocessor.The contents of this dissertation are divided into two chapters.In Chapter One, we mainly consider the upstream weighted alternating segmentexplicit implicit method for the initial and boundary problem of one-dimensionalBurgers equation as below.whereε> 0.Firstly, the grid points on the same time level are divided into several segment.On the di?erent segments of the odd and even time levels , segment implicit scheme explicit scheme . . . explicit scheme segment implicit scheme$and explicitscheme segment implicit scheme . . . segment implicit scheme explicit schemeare used in turn. And segment implicit scheme is that two points on theends of the segment are computed by Saul'yev type asymmetric schemes, and theother points by implicit scheme. Explicit scheme is that all points on the segmentis computed by explicit scheme. According to the above mode, explicit schemeand implicit scheme, two Saul'yev type asymmetric schemes are carried throughalternately.Secondly, by analysis of linearization procedure, prove the stability of the up-stream weighted alternating segment explicit implicit scheme. And the result showsthat the method is absolutely stable.And more, we give some numerical results toshow that the method has good stability and accuracy.In Chapter Two, we mainly consider the upstream weighted alternating blockexplicit implicit method for the initial and boundary problem of two-dimensionalBurgers equation as below.Firstly, we give several block structuresCE block, GE block, IB block, asFig2.1. And the grid points on the same time level are divided into several blocks, on the odd and even time levels , arrange these blocks as Fig 2.2 and Fig 2.3.According to the above mode, explicit scheme and implicit scheme, four Saul'yevtype asymmetric schemes are carried through alternately.Secondly, by analysis of linearization procedure, prove the stability of the up-stream weighted alternating block explicit implicit scheme. And the result showsthat the method is of linear stability. In the end, we give some numerical results toshow that the method has good stability and accuracy.