JACOBI-Spherical Harmonic Approximation And Its Applications For Solving NAVIER-STOKES Equations

It is important and interesting to simulate numerically the movement of fluid flow with spherical geometry: when we study some problems such as weather prediction, fluid flow inside the earth, and some equations in oceanic science and astronomical physics. Since finite difference method is easier for constructing schemes and finite element method is suitable for complex shapes of domainss. we usually used these two methods for numerical simulations of fluid flows. But both of them also have some shortcomings. Firstly: the numerical accuracy is limited by the scheme itself. It means that the numerical error does not. decay, even if the exact solution becomes smoother. Secondly, we need approximate boundary conditions oftentimes. This induces additional errors.Spectral method is one of three basic numerical methods for solving partial differential equations. In the past three decades, more and more attention has been paid to spectral method, along with the rapid development of modern computers. The main advantage of spectral method is its high accuracy. Therefore, it is reasonable to use it for numerical simulation of fluid flow in a ball. But the usual spectral method is only available for periodic or semi-periodic problems, and some problems on rectangular domains. So far, there is no results on spectral method for problems in a ball.During 1990s. Professor Guo Ben-yu etc. developed orthogonal approximation on a spherical surface by taking spherical harmonic functions as base functions. It is very suitable for partial differential equations on a spherical surface. As we know, mixed approximation are oftentimes considered in spectral method and its applications. It is natural to use the spherical coordinates when we consider a problem in a ball. In this case. we take the spherical harmonic functions as base functions on the spherical surface and other systems of orthogonal functions as base functions in the direction of radius. However. the use of spherical coordinates brings new certain difficuhy of numerical simulation, since partial differential equations might possess singularity at the center of ball. Accordingly. the Legendre and Chebyshev spectral approximations are no long available.Recently. Professor Guo Ben-yu eto. established a series of results on the Jacobi approximation, and proposed the Jacobi spectral method for singular problems, which produces the possibility of using spectral mcthod for problems in a ball.In this paper, we establish some basic resul(?)s on mixed Jacobi-spherical harmonic approximation, which serves as the marhematical foundationof mixed spectral method in a ball. We also use this method for the Navier-Stokes equations in a unit ball. The paper consists of four parts. In Chapter 1. we recall briefly the history of related work. present the motivation of this research, and describe the outline of this thesis. In Chapter 2. we establish a series of results on the mixed Jacobi-spherical harmonic approximation. In Chapter 3. we focus on semi discrete Jacobi-spherical harmonic spectral method for solving Navier-Stokes equations with artificial compression in a unit ball. Chapter 4 is devoted to fully discrete Jacobi-spherical harmonic spectral method for solving Navier-Stokes equations with artificial compression in a unit ball.The proposed mixed spectral method for Navier-Stokes equations has several advantages. Firstly, we use the mixed coordinates. More precisely, we use the spherical coordinates for independent variables in space, and so avoid the approximation of boundary conditions on the surface, which is required, if we use the Descartes coordinates. Next, we use the Descartes coordinates for the components of velocity, which simplifies the actual computation and numerical analysis essentially. Thirdly, we adopt the Jacobi approximation in the radial direction and thus overcome the difficulty of dealing with the singularity of equations at the center of ball. which is caused by the spherical coordinates. Furthermore. due to the orthogonality of spherical harmonic functions and Jacobi polynomials, we derive the corresponding discrete systems for the coefficients of expansions of unknown velocity and pressure respectively, which are very suitable for parallel computation. Thereby we save a tot of work. Moreover, we construct the mixed spectral schemes with artificial compression, and so do not impose any artificial boundary condition on the pressure. Finally, the proposed schemes possess generalized stability and spectral accuracy in space. Numerical results demonstrate that our schemes provide accurate numerical solutions even for small modes.The results on the related mixed approximations and techniques developed in this work are also applicable to many other problems with spherical geometry.