# Finite Element Methods for Solving Hydrodynamic Dispersion Equations

• Ne-Zheng Sun

## Abstract

Consider the following two-dimensional advection-dispersion equation
$$L(c) \equiv \frac{{\partial C}}{{\partial t}} + {V_x}\frac{{\partial C}}{{\partial x}} + {V_y}\frac{{\partial C}}{{\partial y}} - \frac{\partial }{{\partial x}}({D_{xy}}\frac{{\partial C}}{{\partial x}} + {D_{xy}}\frac{{\partial C}}{{\partial y}}) - \frac{\partial }{{\partial y}}({D_{xy}}\frac{{\partial C}}{{\partial x}} + {D_{yy}}\frac{{\partial C}}{{\partial y}}) + QC - I = 0,$$
(5.1.1)
which is subject to the initial condition
$$C(x,y,0) = f,{\kern 1pt} (x,y) \in (R),$$
(5.1.2)
boundary conditions
$$C(x,y,t) = {g_1},{\kern 1pt} (x,y) \in ({\Gamma _1}),$$
(5.1.3)
and
$$({D_{xx}}\frac{{\partial C}}{{\partial x}} + {D_{xy}}\frac{{\partial C}}{{\partial y}}){n_x} + ({D_{yx}}\frac{{\partial C}}{{\partial x}} + {D_{yy}}\frac{{\partial C}}{{\partial y}}){n_y} = - {g_2}(x,y) \in ({\Gamma _2}),$$
(5.1.4)
where (R) is the flow domain, (Γ1) and (Γ2) are boundary sections of (R), f is a given function in (R), g 1 and g 2 are given functions along (Γ1) and (Γ2), respectively, and n x and n y are components of the unit outer normal vector to the boundary (Γ2). Equation (5.1.3) expresses the boundary condition of given concentration, i.e., the first-type boundary condition, while Eq. (5.1.4) expresses the boundary condition of given dispersion flux, i.e., the second-type boundary condition.

## Keywords

Finite Element Method Basis Function Local Coordinate System Triangular Element Finite Element Equation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Authors and Affiliations

• Ne-Zheng Sun
• 1
1. 1.Civil and Environmental Engineering DepartmentUniversity of CaliforniaLos AngelesUSA