Some simulations, such as global MHD simulation, space weather forecasting simulation, etc. combine observational data in the model and use it as a boundary condition. Then the boundary condition will change with time. The treatment of such boundary is more complicated as an additional interface between observation data and program is needed. At this time, the boundary condition can be expressed as the following form
where f(t) is decided by the observational data or other time varying data.
Despite its complexity, a time varing boundary condition is a more realistic way to do space numerical simulations.