Editing Finite volume method (section)

==Example==

Consider a simple 1D [[advection]] problem:

{{NumBlk|:|<math>\frac{\partial\rho}{\partial t}+\frac{\partial f}{\partial x}=0,\quad t\ge0.</math>|{{EquationRef|1}}}}

Here, <math> \rho=\rho \left( x,t \right) </math> represents the state variable and <math> f=f \left( \rho \left( x,t \right) \right) </math> represents the [[flux]] or flow of <math> \rho </math>. Conventionally, positive <math> f </math> represents flow to the right while negative <math> f </math> represents flow to the left. If we assume that equation ({{EquationNote|1}}) represents a flowing medium of constant area, we can sub-divide the spatial domain, <math> x </math>, into ''finite volumes'' or ''cells'' with cell centers indexed as <math> i </math>. For a particular cell, <math> i </math>, we can define the ''volume average'' value of <math> {\rho }_i \left( t \right) = \rho \left( x, t \right) </math> at time <math> {t=t_1} </math> and <math>{ x \in \left[ x_{i-1/2} , x_{i+1/2} \right] } </math>, as

{{NumBlk|:|<math>\bar{\rho}_i \left( t_1 \right) = \frac{1}{ x_{i+1/2} - x_{i-1/2}} \int_{x_{i-1/2}}^{x_{i+1/2}} \rho \left(x,t_1 \right)\, dx ,</math>|{{EquationRef|2}}}}

and at time <math> t = t_2 </math> as,

{{NumBlk|:|<math>\bar{\rho}_i \left( t_2 \right) = \frac{1}{x_{i+1/2} - x_{i-1/2}} \int_{x_{i-1/2}}^{x_{i+1/2}} \rho \left(x,t_2 \right)\, dx ,</math>|{{EquationRef|3}}}}

where <math> x_{i-1/2} </math> and <math> x_{i+1/2} </math> represent locations of the upstream and downstream faces or edges respectively of the <math> i^\text{th} </math> cell.

Integrating equation ({{EquationNote|1}}) in time, we have:

{{NumBlk|:|<math>\rho \left( x, t_2 \right) = \rho \left( x, t_1 \right) - \int_{t_1}^{t_2} f_x \left( x,t \right)\, dt,</math>|{{EquationRef|4}}}}

where <math>f_x=\frac{\partial f}{\partial x}</math>.

To obtain the volume average of <math> \rho\left(x,t\right) </math> at time <math> t=t_{2} </math>, we integrate <math> \rho\left(x,t_2 \right) </math> over the cell volume, <math>\left[ x_{i-1/2} , x_{i+1/2} \right] </math> and divide the result by <math>\Delta x_i = x_{i+1/2}-x_{i-1/2} </math>, i.e.

{{NumBlk|:|<math> \bar{\rho}_{i}\left( t_{2}\right) =\frac{1}{\Delta x_i}\int_{x_{i-1/2}}^{x_{i+1/2}}\left\{ \rho\left( x,t_{1}\right) - \int_{t_{1}}^{t_2} f_{x} \left( x,t \right) dt \right\} dx.</math>|{{EquationRef|5}}}}

We assume that <math> f \ </math> is well behaved and that we can reverse the order of integration. Also, recall that flow is normal to the unit area of the cell. Now, since in one dimension <math>f_x \triangleq \nabla \cdot f </math>, we can apply the [[divergence theorem]], i.e. <math>\oint_{v}\nabla\cdot fdv=\oint_{S}f\, dS </math>, and substitute for the volume integral of the [[divergence]] with the values of <math>f(x) </math> evaluated at the cell surface (edges <math>x_{i-1/2} </math> and <math> x_{i+1/2} </math>) of the finite volume as follows:

{{NumBlk|:|<math> \bar{\rho}_i \left( t_2 \right) = \bar{\rho}_i \left( t_1 \right)

- \frac{1}{\Delta x_{i}} 
  \left( \int_{t_1}^{t_2} f_{i + 1/2} dt
- \int_{t_1}^{t_2} f_{i - 1/2} dt
\right) .</math>|{{EquationRef|6}}}}

where <math>f_{i \pm 1/2} =f \left( x_{i \pm 1/2}, t \right) </math>.

We can therefore derive a ''semi-discrete'' numerical scheme for the above problem with cell centers indexed as <math> i </math>, and with cell edge fluxes indexed as <math> i\pm1/2 </math>, by differentiating ({{EquationNote|6}}) with respect to time to obtain:

{{NumBlk|:|<math> \frac{d \bar{\rho}_i}{d t} + \frac{1}{\Delta x_i} \left[ 
f_{i + 1/2} - f_{i - 1/2}  \right] =0 ,</math>|{{EquationRef|7}}}}

where values for the edge fluxes, <math> f_{i \pm 1/2} </math>, can be reconstructed by [[interpolation]] or [[extrapolation]] of the cell averages. Equation ({{EquationNote|7}}) is ''exact'' for the volume averages; i.e., no approximations have been made during its derivation.

This method can also be applied to a [[Finite volume method for two dimensional diffusion problem|2D]] situation by considering the north and south faces along with the east and west faces around a node.