Finite Volume Element Methods For Some Wave Equations In Fluid Dynamics

The finite volume element method has been an important numerical tool for solv-ing partial differential equations. Since the method is easily implemented, can effi-ciently deal with complex geometries and naturally preserve main physical conserva-tion laws, it has attracted considerable interest. This work is devoted to finite volume element methods for some wave equations in fluid dynamics. We design some effec-tive finite volume element schemes to study these wave equations.Firstly, we consider the hyperbolic partial differential-difference equation with shift and design standard finite volume scheme and upwind finite volume element scheme for solving it. At the same time, L2error estimate is given for upwind finite volume element scheme. Several numerical experiments are performed to check the efficiency and convergence of the numerical schemes.Secondly, we study the stochastic damped improved Boussinesq equation. To numerically solve the equation, we use quadratic Lagrange functions to approximate space derivative, three-order strong-stability-preserving scheme to discretize time derivative and Monte Carlo method to deal with stochastic term. We have obtained fully discrete finite volume element scheme for the equation. By the scheme, we nu-merical investigate the influence of a noise term on mass of system and amplitude of solitary wave.Next, we consider the two-dimensional quasi-geostrophic equations on a sphere and propose a new Fourier finite volume element method. By using piecewise linear functions in the latitudinal direction, Fourier discretization in the longitudinal direc-tion and leap-frog scheme to discretize time derivative, we can get the Fourier finite volume element schemes. Some numerical results illustrate that the new method is second-order convergence, and can conserve the energy and enstrophy of system and overcome pole problem.At last, we discuss optimal control of air quality based on derivative-free opti-mization method. The characteristic finite difference method is used to solving the air pollution model which describe the development of pollutant. The locations of industrial pollutant sources are chosen as the decision variables. By defining rel- ative objective functions, we solve the optimization problems using derivative-free optimization method. Some numerical examples demonstrate that derivative-free op-timization method can be used to solve these problems. The optimal locations of pollution sources can meet the air quality index at low cost.