Local Discontinuous Finite Element Method For Air Pollution Model And Heat Conduction Equation

People build air pollution models to monitor and predict the concentration of pol-lutants,so as to control air pollution.Therefore,it is very important for us to solve the air pollution models.Most of air pollution models are partial differential equation-s.In recent years,the local discontinuous Galerkin(LDG)method has widely used for solving partial differential equations.The LDG method has many good properties:high convergence accuracy and strong stability and so forth.We build the local discontinuous Petrov-Galerkin(LDPG)method on the basis of the LDG method.The LDPG method not only has the advantages of the LDG method,but also simplifies the calculation pro-cess.In this article,we established the LDG and LDPG numerical formats of the air pollution model,and gave the numerical simulation results.At the same time,we stud-ied the LDPG scheme of the two-dimensional heat conduction equation and gave the numerical simulation results.The main results of this paper have three parts.Firstly,we construct the LDG scheme to solve two class of air pollution models.The numerical flux in the LDG scheme selects the”alternating flux”and the Lax-Friedrichs flux.The time discretization uses the third-order TVD Rnuge-Kutta scheme.Furthermore,a numerical example is given to confirm the effectiveness of the method.Secondly,we construct the LDPG scheme of air pollution models.The LDPG method can select different test function spaces and trial function spaces.Numerical examples show that when the basis function is selected as the quadratic element,the method has third-order accuracy.In the meantime,we analyzed and compared the LDPG method with the finite volume element method and the finite difference method.Further,we also analyzed the two methods,LDG and LDPG,and discussed their advantages and disadvantages.Finally,on the basis of rectangular partition,the trial function space is taken as a function space formed by a piecewise m-degree polynomial.And we set up the test function space on the basis of the dual partition.Therefore,the LDPG scheme is established to solve two dimensional heat conduction equation.Then the numerical simulation is carried out,and the convergence order in L¹ and L^? norms can get up to(?^K+1).