Lattice Boltzmann Models For Coupled Convection-diffusion Equations And Its Applications

The convection-diffusion equation is a mathematical model for describing the phe-nomenon of convection and diffusion, which has a wide range of applications in natural and engineering areas, such as pollution control, oil transportation, chemical reaction, dis-ease treatment and micro fluidic control. However, the actual problem is often described by coupled equations with variable convection term, which makes it difficult to obtain exact so-lutions for general coupled convection-diffusion equations. With the rapid development of computer technology, numerical method has become an effective tool to solve those com-plex problems. With the characteristics of parallel nature, simple and efficient algorithm, common format, a new mesoscopic lattice Boltzmann method is also used to study the cou-pled convection-diffusion equation.The influence of variable coefficient of convection term cannot always be neglected in the study of single or coupled convection-diffusion equations. In this paper, we give two lattice Boltzmann models for steady or unsteady convection-diffusion equation with vari-able convection term, respectively. And then we propose several lattice Boltzmann models for the conservative and nonconservative form of coupled convection-diffusion equations through the effect of the source term. Next, a class of generalized viscous wave equation with third order mixed partial derivatives of time and space is solved by transform into the special coupled equations. At last, spiral wave dynamic described by the coupled equa-tions and Bose-Einstein condensation controlled by the complex equation are carried on the preliminary exploration.The main work have been carried out from the lattice Boltzmann models establishment and the applied research. In terms of the lattice Boltzmann models establishment, four aspects of the research have been carried out as follows:(1) An efficient lattice Boltzmann model for n-dimensional steady convection-diffusion equation with variable coefficients in the convection term is proposed through modifying the equilibrium distribution function properly. The model has some distinct characteristics, such as simple in calculation format and fast convergence rate.(2) A lattice Boltzmann model for a class of n-dimensional unsteady convection-diffusion equations with variable coefficients is proposed through introducing an auxiliary distribution function. The LBGK models for convection-diffusion equation with constant variable is included in the present model, which is an extension of the previous models.(3) Two lattice Boltzmann models for the conservative form of coupled convection- diffusion equations and coupled viscous Burgers equations are proposed through the special handling of the source terms and the character of the one dimensional equations, respec-tively. Two lattice Boltzmann models for the nonconservative form of 2-dimensional cou-pled Burgers equations are proposed through the source term and character of the lattice Boltzmann method. Both Chapman-Enskog analysis and numerical simulations show the effectiveness of the coupled models.(4) Based on the research of the coupled convection-diffusion equations, a class of generalized viscous wave equation with third order mixed partial derivatives of time and space is also solved by transforming into the special coupled equations, and solved later by the lattice Boltzmann method. Numerical studies show that the lattice Boltzmann method can solve this kind of equation effectively.Based on the above models, two aspects in the following are carried out in this thesis:(1) The research on spiral waves is of great significance in the disease caused by tachy-cardia and atrial fibrillation. The proposed models for coupled convection-diffusion equa-tions is used to simulates the spiral wave form, disappearance in the electric field, deforma-tion and breakup with straining.(2) The proposed models for convection-diffusion equation with variable coefficient in convection term is used to simulate the focusing and defocusing phenomena in Bose-Einstein condensation. The numerical results show that the lattice Boltzmann method can be effectively used in the study of the Bose-Einstein condensation, which expands the ap-plication range of the lattice Boltzmann method.In summary, this thesis have conducted a comprensive study on the single and coupled convection-diffusion equations from theoretical models and practical applications. In view of the charastic of the convection-diffusion equations, we propose several lattice Boltzmann models for single and coupled convection-diffusion equations through treatments of the source term. Besides, the model is validated by the means of the effectiveness through the Chapman-Enskog analysis and numerical simulation. Furthermore, we make a preliminary study on spiral waves and promising Bose-Einstein condensation problems, which is described by coupled convection-diffusion equations. This work is a necessary theoretical and practical basis for the future studies.