NumLib::FullUpwind Class Reference

Detailed Description

For the full upwind scheme, we consider the general advection term of the advection diffusion transport equation:

$$\nabla \cdot ( u \mathbf{v}),$$

where $u$ can be temperature $T$ or mass component concentration $C$, and $\mathbf{v}$ is the fluid velocity. This means $\nabla \cdot \mathbf{v}$ may not be zero, or physically the fluid flow may be compressible.

The discretized weak form of that advection term takes the form

$$\int_{\Omega_e} \nabla \cdot ( u \mathbf{v}) \phi_i \mathrm{d} \Omega = \int_{\Omega_e} \nabla \cdot ( u \mathbf{v} \phi_i) \mathrm{d} \Omega - \int_{\Omega_e} \nabla \phi_i \cdot ( u \mathbf{v} ) \mathrm{d} \Omega$$

with $\phi_i$ the test function. The first term on the right hand side can be converted to boundary integration according to the Green's theorem as

$$\int_{\Omega_e} \nabla \cdot ( u \mathbf{v} \phi_i) \mathrm{d} \Omega = \int_{\partial\Omega_e} ( u \mathbf{v} \phi_i) \cdot \mathbf{n} \mathrm{d} \Gamma$$

with $\mathbf{n}$ the normal to the boundary surface. That boundary integration is a part of Neumann boundary condition. Therefore what left for the element integration is

$$-\int_{\Omega_e} \nabla \phi_i \cdot ( u \mathbf{v} ) \mathrm{d} \Omega,$$

which is denoted as $R_i$ hereafter.

Based on the scheme introduced by Dalen [10], the full upwind scheme evaluates the following flux related quantity for each node

$$q_i = -\int_{\Omega_e} \nabla \phi_i \cdot \mathbf{v} \mathrm{d} \Omega$$

to determine whether it is an upwind node. If $q_i>0$, node $i$ is at upwind position.

Let

$${q}_{up}=\sum_{q_i \geq 0} q_i u_i$$

be the total flux at the upwind nodes, and

$${q}_{down}=-\sum_{q_i < 0} q_i$$

be the total flux related quantity at the down nodes, we can approximate the discretized weak form of the advection term as:

$$\begin{eqnarray*} R_i & \approx& \begin{cases} q_i\,u_i,\forall q_i \geq 0\\ q_i \frac{{q}_{up}}{{q}_{down}},\forall q_i < 0 \end{cases}\\ & = & \begin{cases} q_i\,u_i,\forall q_i \geq 0\\ \frac{1}{{q}_{down}} q_i {\sum_{q_j \geq 0} (q_j u_j)},\forall q_i < 0 \end{cases} =\tilde{R}_i.\\ \end{eqnarray*}$$

The above approximation defines the full upwind scheme of the advection term of the advection diffusion transport equation for the FEM analysis. Obviously, we see that

$$\begin{eqnarray*} \sum_i \tilde{R}_i &=& \sum_{q_i \geq 0} q_i u_i + \sum_{q_i < 0} \frac{1}{{q}_{down}} q_i {\sum_{q_j \geq 0} (q_j u_j)}\\ &=& \sum_{q_i \geq 0} q_i u_i + \frac{\sum_{q_i < 0} q_i}{{q}_{down}} {\sum_{q_j \geq 0} (q_j u_j)} = q_{up}-q_{up} = 0, \end{eqnarray*}$$

which means the nodal mass balance of element is guaranteed.

Definition at line 160 of file NumericalStabilization.h.

#include <NumericalStabilization.h>

Public Member Functions
	FullUpwind (double const cutoff_velocity)
double	getCutoffVelocity () const

Private Attributes
double const	cutoff_velocity_

Constructor & Destructor Documentation

FullUpwind()

NumLib::FullUpwind::FullUpwind ( double const cutoff_velocity )

inlineexplicit

Definition at line 163 of file NumericalStabilization.h.

        : cutoff_velocity_(cutoff_velocity)
    {
    }

Member Function Documentation

getCutoffVelocity()

double NumLib::FullUpwind::getCutoffVelocity ( ) const

inline

Definition at line 168 of file NumericalStabilization.h.

{ return cutoff_velocity_; }

References cutoff_velocity_.

Member Data Documentation

cutoff_velocity_

double const NumLib::FullUpwind::cutoff_velocity_

private

The cutoff velocity. The stabilization is not applied if the velocity magnitude is below the cutoff velocity.

Definition at line 173 of file NumericalStabilization.h.

Referenced by getCutoffVelocity().

---

The documentation for this class was generated from the following file:

• NumLib/NumericalStability/NumericalStabilization.h