Interior Flux

The boundary condition can be used on interior boundaries only. With this node, the boundary condition is enforced according to:

where g can be specified. Here, n is outward normal seen from the downside, so it should be read as −n · Γ = g for a downside of an interior boundary and as −n · Γ = −g on the upside of an interior boundary. If a dependent variable is used in this expression (without up or down operators), an implicit mean operation is invoked taking the average of the up and down values.

To appreciate how this boundary condition works, consider a simple example of a one-way transport equation in 1D

with appropriate initial data and boundary conditions, and where the parameter a > 0 jumps (it is a discontinuous function of x) at an internal interface.

The proper upwind numerical flux condition is not obtained by using the internal Lax-Friedrichs flux. It can be shown that this is obtained by the numerical flux

where r and l denote the right and left side of the interface, respectively.

Since the downside coincides with the left side in 1D (n = 1), this condition can be expressed by setting

g=-2*down(a)*up(a)*down(u)/(down(a)+up(a))

In general and in higher dimensions, one typically needs the down (and up) versions of the mesh normal to express these conditions. For example, when the sign of the normal is unknown (that is, which side of an interface COMSOL Multiphysics considers the upside an downside), the above condition can be entered as

g = -down(a)*up(a)/(down(a)+up(a))*dnx*(up(u)+down(u)+dnx*(down(u)-up(u)))

Here dnx means the downside normal (dnx=-unx).

It is important to use the mesh version of this vector in higher dimensions on curved boundaries. For example, dnxmesh, dnymesh, and dnzmesh denotes the x, y, and z components of the mesh normal vector on the downside.

Boundary Selection

Only interior boundaries can be selected. See Working with Geometric Entities.

Boundary Flux/Source

Enter the flux term g (SI unit: 1/m).