An Integral Equation Approach To The Convection Diffusion Equations And Navier-Stokes Equations

The convection-diffusion equation and Navier-Stokes equation are two basic equations in fluid dynamics. They have been widely applied in physics, chemistry and engineering. The analytical solutions of these equations are only available in some simple cases. Usually, it is difficult to get analytical solutions of the convection-diffusion equation and Navier-Stokes equation. Thus, numerical method becomes a powerful tool for solving these equations in domains with complex geometies. For the convection dominated convection-diffusion problems and the complex flows with high Reynolds number, it is of great value to construct accurate, stable and efficient numerical methods. Recently, the meshless methods, especially those based on the integral equation theory have been gaining interest for its high accuracy and reduction of computational costs. Encouraged by the successful use of these methods, in the present work, we attempt to propose an integral equation approach for solving the convection-diffusion equation and Navier-Stokes equation.When studying the steady state convection-diffusion equation, a Laplace equation about the Green’s function is firstly introduced. Then the Laplace equation is discretized into algebraic equations by expanding the Green’s function into series form. Solving this algebraic equation system and using the property of the Green’s function, the convection-diffusion equation can be transformed into integral equation. Expanding the unknown function with the same series and using its orthogonality, the integral equation is reduced into an algebraic equation system. Solving this algebraic equation system, we can obtain the approximate solution of the convection-diffusion equation. With the help of Chebyshev polynomial and Fourier series, the integral equation approach is used to approximate the one-dimensional convection-diffusion problems with nonhomogeneous boundary conditions and the two-dimensional convection-diffusion problems with homogeneous boundary conditions. The comparisons with the finite volume method, the finite element method and the upwind difference method show that the integral equation approach is more accurate and stable. The good stability is also illustrated by the convection-dominated convection-diffusion problems.In the unsteady state convection-diffusion equation, the time variable appears, so we need to consider the time discretization method. Here, we choose the Crank-Nicolson method for the time discretization, as it can not only give a simple discrete scheme, but also give rise to a good accuracy. Then by using the Green’s function of the Laplace equation in the series form, the convection-diffusion equation is transformed into an integral equation that is further converted into an algebraic equation system. Solving this algebraic equation system, we can get the approximate solution of the original problem in the series form. In the section of numerical experiments, two one-dimensional convection-diffusion problems and four two-dimensional convection-diffusion problems are used to examine this integral equation approach. For one-dimensional problem the Dirichlet boundary condition is prescribed on one side and the Neumann boundary condition on the other. In the two-dimensional cases, both convection-dominated convection-diffusion problem and convection-diffusion problem with non-constant convection velocity are considered. In comparison with the characteristic variational multiscale method, the integral equation approach is more efficient. And both methods can give rise to numerical solutions with a high accuracy even with small grid numbers. The comparison between the integral equation approach and the finite volume element method shows that the integral equation approach doesn’t have obvious advantage for the small number of grid points, but as increas’ng the number of the grid points and decreasing the time step, the integral eqation approach demosntrates a better accuacry. Furhermore, the convergence speed of the integral equation approach is much faster than that of the finite volume element method.The velocity-pressure linkage is one of the main difficulties for solving Navier-Stokes equation. Several mehtods are avaiable for dealing with this problem. Here, we choose the projection method to solve this problem by considering the feature of the integral equation approach. According to the form of the discrete equations given by the preject method, we give different Laplace equations about the Green’s function. By using the property of the Green’s function, the Navier-Stokes equation can be converted into integral equations. As the boundary condition of the Navier-Stokes equation is in the general form and the intermediate variable of the projection method gives a Neumann boundary condition, we expand the Green’s functions and the unknown functions by using the Chebyshev polynomial. With the help of the property of the Chebyshev polynomial, we can get three algebraic equation systems about the unknown functions. Solving these algebraic equation systems, we can obtain the numerical solution of the Navier-Stokes equation. Finally, the accuracy and performance of the integral equation approach are examined by solving an unsteady Navier-Stokes equation. In comparison with the fractional-step method it leads to more accurate results and takes less CPU-time.In this thesis we also investigate the convection-diffusion equation and Navier-Stokes equation with period boundary conditions. In this case, the Green’s function and the unknown function can be expanded in Fourier series form, so the discretization scheme is simpler than that for other boundary conditions. We also need to construct a Laplace equation about the Green’s function, and transform the convection-diffusion equation or the Navier-Stokes equation into the integral equation which is further reduced to the ordinary differential equations system. Then the TVD Runge-Kutta method is employed to numerically solve the equations system. In comparison with the local discontinuous Galerkin method the integral equation approach shows a better convergence, especially for the convection dominated and nonlinear convection-diffusion problems. Furthermore, the results given by the integral equation approach is much more accurate than that obtained by the projection method in simulating the Taylor vortices. Finally, the integral equation approach demonstrates a high accuracy in simulating the incompressible flows with high Reynolds number.