Efficient Spectral Methods For Non-linear Aerosol Dynamic Equations

Aerosol is an important component of the atmosphere.It can affect the various physical and chemical reaction of atmosphere,and then affect the change of the environment.What's more it can directly affect the healthy of human.Therefore,the in-depth study of the behavior of aerosol particles has become one of the important topics in the atmospheric environment.The aerosol dynamic equation describes the functional relationships between the physical chemistry processes,such as nucleation,polymerization,condensation,and sedimentation,with the aerosol particle size.It's a nonlinear integral-differential equation.In this article,we only consider the aerosol dynamics equation which including coagulation,sedimentation and polymerization process.The equation model is as follows:In this thesis,we use spectral methods to solve the aerosol dynamics equation.The spectral method originated from the Ritz-Galerkin method.It is an important and effective numerical method for solving partial differential equations.The advantage of the spectral method is the so-called ”infinite-order convergence”.What's more,the Fast Fourier Transform can greatly improve the computing speed.Using the spectral method to solve the aerosol dynamic equation can not only ensure higher calculation accuracy,but also improve the calculation speed.The essence of the spectral method is to write the function approximately as a finite series expansion of a smooth function.This is similar to the finite element method,the difference is that the expansion of spectral method is based on the whole range of values.It is a overall algorithm.So it ensures the high precision of spectral method.In this paper,we introduced the related knowledge of atmospheric aerosol and its research status at home and abroad.And then we present the equation model of aerosol dynamics equation.And introduced the coagulation,sedimentation and polymerization process of aerosol dynamics equation.Then briefly introduced the knowledge of spectral methods and give the derivation process of using spectral-collocation method and chebyshev-spectral method to solve the aerosol dynamic equation.Finally we do numerical experiments.In the first three examples,we proved the advantages of high accuracy and high speed of spectral methods.The remaining three examples are used to analysis practical problems.