We introduce a new method for solving non-stationary advection equations. The new method is based on finite volume space discretization and a semi-implicit discretization in time. Its basic idea is that outflow from a cell is treated explicitly while inflow is treated implicitly. This is natural, since we know what is outflowing from a cell at the old time step but we leave the method to resolve a system of equations determined by the inflows to a cell to obtain the solution values at the new time step. Our new method is exact for constant velocity transport of quadratic functions in any dimension and for any length of a time step and it is second order accurate for smooth solutions in general. The matrix of the system is determined by the inflow fluxes which results in a M-matrix yielding favourable stability properties for the scheme. The method allows large time steps at a fixed spatial grid without losing stability and not deteriorating precision. This makes the new method attractive for practical applications. The scheme is well suited for variable velocity vector fields in higher dimensions and for nonlinear advection problems which is documented by a series of numerical experiments.

@TechReport{MO10a,
  author       = {Mikula, K. and Ohlberger, M.},
  title        = {A New Inflow-Implicit/Outflow-Explicit Finite Volume Method for Solving Variable Velocity Advection Equations},
  institution  = {FB 10 , Universit\ät M\ünster},
  number       = {01/10 - N},
  month        = {june},
  year         = {2010},
  type         = {Preprint},
  url          = \{/2010/MO10a},
}