Second order accurate FV discretization for a class of interface
        problems with discontinuous solution

T.Chernogorova,
Faculty of Mathematics and Informatics, University of Sofia

O.Iliev,
Institute of Mathematics and Informatics,
Bulgarian Academy of Science


Problems with discontinuous coefficients (so called
interface problems) are considered in the case when the solution
is discontinuous, and the flux is continuous on the
interfaces. These problems are known also as imperfect contact
problems. Finite volume discretizations of such problems on cell
centred grids are investigated. A new scheme is derived under the
following assumptions: i) the interfaces are aligned with the grid
cells faces; ii) the diffusivity coefficient is a constant within any
grid cell; iii) the flux is continuously differentiable on the
interfaces. The scheme exploits minimal stencil: 3 points in 1D,
5 points in 2D. The fluxes on interfaces are approximated with
third order. Pointwise second order convergence for the new scheme is
theoretically proved, and numerically confirmed. Results from computations
with known finite volume discretizations are also presented in order to
demonstrate the advantage of the new scheme.