Inflow-Implicit/Outflow-Explicit Scheme for Solving Advection Equations

We present new method for solving non-stationary advection equations based on the finite volume space discretization and the semi-implicit discretization in time. Its basic idea is that outflow from a cell is treated explicitly while inflow is treated implicitly. Since the matrix of the system in this new I2OE method is determined by the inflow fluxes it is an M-matrix yielding favourable solvability and stability properties. The method allows large time steps at a fixed spatial grid without losing stability and not deteriorating precision which makes it attractive for practical applications. Our newmethod is exact for any choice of a discrete time step on uniform rectangular grids in the case of constant velocity transport of quadratic functions in any dimension. We show that it is formally second order accurate in space and time for 1D advection problems with variable velocity and numerical experiments indicates its second order accuracy for smooth solutions in general.