High-order characteristic-finite volume methods for aerosol dynamic equations

Abstract

Aerosol particles have an important effect on changing of climate and human health, where aerosols scatter and absorb the incoming solar radiation, and thus decrease the precipitation efficiency of warm clouds and can cause an indirect radiative forcing associated with changes in cloud properties. Meanwhile, it has also been recognized that the particles of aerosols in the sub-micrometer size range can be inhaled and thus pose certain health hazards. In this paper we analyze the finite volume method based on linear interpolation and Hermite interpolation combined with the method of characteristics for the nonlinear aerosol dynamic equations on time and particle size, which involve the advection process, condensation process and the nonlinear coagulation process. Numerical experiments for the multiple log-normal aerosol distributions are further given to confirm the theoretical results.

Introduction

Aerosol particles have an important effect on change of climate and human health, where aerosols scatter and absorb the incoming solar radiation, and thus decrease the precipitation efficiency of warm clouds and can cause an indirect radiative forcing associated with changes in cloud properties. Meanwhile, it has also been recognized that the particles of aerosols in the sub-micrometer size range can be inhaled and thus pose certain health hazards. Therefore, the study of prediction of aerosol distributions of different chemical and dynamic processes has been an important topic. Furthermore, aerosol modeling [1] has been playing a significant role in terms of studying and simulating the behavior of aerosol dynamics in environment.

The aerosol dynamic equations in terms of the aerosol size distribution function describe different processes evolved in the lifetime of aerosols, which include condensation, nucleation, coagulation, and deposition. In recent years, many numerical methods have been studied to solve the aerosol dynamic equations, such as sectional method [2], moment method [3], [4], [5], [6], [7], [8], [9], stochastic approach [10], maximum entropy method [11]. The moment method tends to be the chemical or physical method with single modal distribution based on the aerosol physical properties but is not suitable to practical multi-modal distribution processes. Recently, Liang et al. developed a wavelet Galerkin Method [12], second order characteristic finite element method [13], [14], splitting wavelet method [15] and finite volume method [16], etc.

In this paper, we consider the numerical methods for the following aerosol dynamic equations [11], [17], [18]

$$\frac{\partial u(v,t)}{\partial t}=-\frac{\partial \left(G\left(v\right)u(v,t)\right)}{\partial v}+\frac{1}{2}{\int }_{V_{min}}^{v-V_{min}}\beta (v-w,w)u(v-w,t)u(w,t)dw$$$$-u(v,t){\int }_{V_{min}}^{V_{max}}\beta (v,w)u(w,t)dw-R\left(v\right)u(v,t),$$

$$u(V_{min},t)=0,t\in (0,T],$$$$u(v,0)=u_0\left(v\right),v\in [V_{min},V_{max}],$$

where $t>0$ is time, $v$ is the volume of the aerosol particle, $u(v,t)$ is the concentration distribution of the aerosol particle, $T>0$ is the time interval.

In practice, one usually assumes that the aerosol particle distribution has a nonzero minimal volume $V_{min}$ and a finite maximal volume $V_{max}$, which means that the aerosol dynamic equations are built on a finite volume interval $\Omega =[V_{min};V_{max}]$.

The terms of the growth rate $G\left(v\right)$ and the coagulation kernel $\beta (v,w)$ are associated with the atmosphere surroundings. The particle growth rate $G\left(v\right)$ can be expressed as $G\left(v\right)=G_{\gamma }{v}^{\gamma }$, $0<\gamma \le 1$, where $G_{\gamma }$ is a positive number involving numerical and physical constants and difference in vapor pressure of the diffusing species in bulk gas [1]. The coagulation is mostly due to the Brownian activity. The coefficient $\beta (v,w)$ is the coagulation kernel between the aerosol particles of volume $v$ and $w$ entering collision. The coagulation kernel $\beta (v,w)$ is bounded in finite volume, there exists ${\beta }_0$ such that $\beta (v,w)\le {\beta }_0$. The last term on the right side of (1) is associated with the removal process, which describes particles of volume $v$ lost due to sinks of aerosols. $R\left(v\right)$ is the removal rate for a particle of volume v and there exists a positive real number $R_0$such that $R\left(v\right)\le R_0$.

First, we introduce the transformation [12], [15]. Let

$$x=av+b,a=\frac{1}{V_{max}-V_{min}},b=-\frac{V_{min}}{V_{max}-V_{min}}.$$

Thus we can get:

$$\frac{\partial u(x,t)}{\partial t}=-\frac{\partial \left(G\left(x\right)u(x,t)\right)}{\partial x}+{\int }_0^x\beta (x-y,y)u(x-y,t)u(y,t)dy-2u(x,t){\int }_0^1\beta (x,y)u(y,t)dy-R\left(x\right)u(x,t),$$

$$u(V_{min},t)=0,t\in (0,T],$$$$u(x,0)=u_0\left(x\right),x\in \Omega ,$$

where $G\left(x\right)=aG\left(v\right)$,  $\beta (x,y)=\frac{1}{2a}\beta (v,w)$.

For this problem, the finite volume method (FVM) based on the linear interpolation was proposed in [16]. The finite volume method, which is also called the generalized difference method, introduces a volume integral formulation of the differential equation with a finite partitioning set of volume to discrete the solution in [19], [20]. The finite volume has a wide range of applications in scientific and engineering computations (cf., [21], [22], [23], [24], [25]). A typical finite volume method [26], [27], [28] introduces piecewise constant functions as test functions, where it keeps the same dimension for the spaces of the trial functions and test functions, two different partitions of the domain are needed, one called primal partition which is associated with the trial space and one called dual partition which is associated with the test space.

In this paper, we develop and analyze the high-order characteristic finite volume methods for the aerosol dynamics. We first propose the linear characteristic finite volume method and derive its error estimate. Then, we propose and analyze the characteristic Hermite finite volume method for aerosol dynamics.

The paper is organized as follows. In Section 2, the finite volume method based on linear interpolation combined with the method of characteristics is proposed and the associated error estimates are presented. In Section 3, the finite volume method based on Hermite interpolation combined with the method of characteristics is proposed and the associated error estimates are presented. In Section 4, numerical experiments for multiple log-normal distributions are further given.