A finite volume local defect correction method for solving the transport equation

The local defect correction (LDC) method is applied in combination with standard finite volume discretizations to solve the advection-diffusion equation for a passive tracer. The solution is computed on a composite grid, i.e. a union of a global coarse grid and local fine grids. For the test a dipole colliding with a no-slip wall is used to provide an actively changing velocity field. The LDC method is tested for the problem of localized patch of tracer material that is transported by the provided velocity field. The LDC algorithm can be formulated to conserve the total amount of tracer material. However, if the local fine grids are moved to adaptively follow the behaviour of the solution, a loss or gain in the total amount of tracer material is produced. This deficit in tracer material is created when the solution is interpolated to obtain data for the moved fine grid. The data obtained by the interpolation scheme in the new refined region can be adapted in such a way that the deficit is spread over the new grid points and conservation of tracer material is satisfied. Finally, the results of the conservative finite volume LDC method are compared and validated with results from a spectral method.