## 3.2. Numerics

If the case is transient rotor-stator, then the current relative position of each side of a sliding interface is first computed at the start of each timestep. Each frame change model then proceeds by discretizing the surface fluxes along each side of the interface in terms of nodal dependent variables, and in terms of control surface equations and control surface variables. Each interface surface flow is discretized using the standard flux discretization approach, but employs both the nodal dependent variables on the local side of the interface and control surface variables on the interface region. Balance equations within the interface region are generated for the interface variables from the flux contributions from both sides of the interface. These equations are called control surface equations (different from control volume equations) because they enforce a balance of surface flows over a given surface area.

In detail, the GGI connection condition is implemented as follows:

1. Define regions within the interface where the fluxes must balance: control surfaces. Within each control surface, identify new dependent variables. These are called interface variables. For a Stage (Mixing-Plane) interface, the balance is across the entire interface in the direction of rotation, with as many control surfaces perpendicular to the direction of rotation as the grid permits. For all other interfaces, the control surface balance is at the resolution of the interface grid.

2. Evaluate the fluxes at each interface location, by visiting all control volumes with surfaces exposed to the interface. Evaluate the surface flows using the `standard' approach taken for all interior flux evaluations for advection, diffusion, pressure in momentum, and mass flows. Use a combination of nodal dependent variables and the interface variables in these evaluations. For example consider advection; if the flow is into the interface control volume, the advected quantity is equated to the interface variable. If the flow is out of the interface control volume, the advected quantity is equated to the local nodal control volume variable. Below is a summary of all common flux discretizations at the interface:

   - Advection: Mass out is connected to the upstream (nodal) values, and mass in is connected to upstream (control surface) values.

   - Diffusion: A diffusion gradient is estimated using the regular shape function based gradient coefficients, but all dependence of the gradient estimate on nodes on the interface are changed to a dependence on interface variables.

   - Pressure in momentum: Evaluated using local nodal and control surface pressures and shape function interpolations.

   - Local pressure gradient in mass redistribution: This gradient is estimated using the regular shape function based gradient coefficients, but all dependence of the gradient estimate on nodal pressure on the interface is in terms of the interface pressure variable.

3. When a face is in contact with more than one control surface balance equation, discretize the fluxes at each integration point location in terms of generic interface unknowns, evaluate the flux N times (where N is the number of control surfaces in contact with the face), each time using a different control surface variable and applying a weighting factor to the flow based on an `exposed fraction' basis. Each partial flow is accumulated in the control volume equation and in the relevant control surface equation.

4. Include each surface flow evaluation in two places: once in the interface control volume equation, and once in the adjacent control surface equation. Once all interface surfaces have been visited, the resulting equation set is as follows:

   - All interface control volume equations are complete. Each equation has coefficients to the usual neighboring nodal variables, as well as to interface variables.

   - All control surface equations are now complete. Each equation has coefficients to local interface variables as well as to nodal variables.

5. Solve the linear equation set to yield values of all nodal variables (from the control volume equations) and all interface variables (from the control surface equations).

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