Mixed Spectral Method For Exterior Problems Of Navier-Stokes Equations

Spectral methods developed rapidly in the past three decades, which serve asimportant tools for solving di?erential equations numerically. The fascinating meritof spectral methods is their high accuracy. Thereby, they have been applied suc-cessfully to numerical simulations in many fields, such as ?uid dynamics, quantummechanics, earth science and so on. The usual spectral methods are only availablefor periodic problems and some problems defined on bounded rectangular domains.However, many practical problems arising in science and engineering require solvingpartial di?erential equations defined on unbounded domains or exterior problems.One of the methods for solving such problems is to set certain artificial boundaries,impose some artificial boundary conditions, and then resolve the corresponding ap-proximated problems by using finite di?erence method or finite element method.Whereas, these treatments cause additional errors. Thus, it seems better to solvethem directly.The purpose of this work is to develop mixed generalized Laguerre-Fourier spec-tral methods for solving Navier-Stokes equations outside a disc. Of course, we cansolve the primitive equations directly. But in this case, we face to three di?cultproblems. The first one is how to match the velocity and pressure on the boundaryof obstacle exactly. We may deal with the boundary values approximately. However,it induces additional errors. In opposite, if we adopt the polar coordinates, then theabove trouble no longer appears. Next, the underlying problem is defined on an unbounded domain, for which we often set an artificial boundary and impose someartificial boundary conditions. But, this treatment also bring additional numericalerrors. Therefore, we shall use the generalized Laguerre polynomials or functionsto solve such problems properly. The third problem is how to deal with the in-compressibility of numerical solutions in actual computation. For remedying thisdeficiency, we follow the idea of Professor Guo Ben-yu and other authors to designnumerical algorithm based on the stream function form of the Navier-Stokes equa-tions, in which the incompressibility is fulfilled automatically. Whereas, we have toinvestigate the spectral method for nonlinear exterior problems of fourth order.This thesis consists of five chapters. The first chapter is for introduction.The second chapter is for preliminaries. In chapter 3, we investigate the mixedgeneralized Laguerre-Fourier orthogonal aprroximation and its applications toexterior problems of partial di?erential equations of fourth order. Some basicresults are established. As an important application, a mixed spectral scheme isproposed for the stream function form of the Navier-Stokes equations outside adisc. The numerical solution fulfills the compressibility automatically. The stabilityand convergence of the proposed scheme are proved. Numerical results demonstrateits spectral accuracy in space, and coincide with the theoretical analysis well. Inchapter 4, we investigate the mixed Laguerre-Fourier orthogonal approximation byusing generalized Laguerre functions, with its applications to exterior problems ofpartial di?erential equations of fourth order. Some basic results are established.The advantage of this approximation is that the factor e?βρdoes no longer appearin the weight function. So it is more natural and simplifies theoretical analysis and actual computation. The corresponding mixed spectral scheme is providedfor the stream function form of the Navier-Stokes equations outside a disc. Thenumerical solution also fulfills the compressibility automatically. The stability andconvergence of the proposed scheme are analyzed. Numerical results demonstrateits spectral accuracy in space, and confirm the analysis well. The final chapter isfor some concluding discussion.