A second order accurate, space-time limited, bdf scheme for the linear advection equation

Steady supersonic flow fields are frequently calculated by solution of the Euler or Parabolized Navier-Stokes (PNS) equations via a space-marching algorithm. Within space-marching the streamwise direction is treated in a time-like manner and the ensuing discretization allows solution of a 3 dimensional problem as a sequence of 2-D problems leading to high efficiency. The associated finite-volume discretizations are cell-centred in the crossflow direction and coincide with the mesh in the time-like space-marching direction. An alternative approach is the locally iterated method (Newsome et al., 1987) in which a plane-by-plane relaxation of the supersonic Euler or PNS equations is performed on a mesh centred in all 3 coordinates. Second order accuracy may be sought using an unlimited extrapolation procedure in the streamwise (time-like) direction. In this work we consider the model problem of 1-dimensional linear advection. As noted previously (Thompson and Matus, 1989) we show that the above mentioned locally iterated methods may be regarded as implicit backward differentiation formulae of second order accuracy. If a purely upwind difference is taken in the streamwise direction then the resulting scheme is stable but dispersive. Such behaviour is explained by regarding components of the BDF time integration as a local extrapolation of the dependent variable in the time-direction and it may be eliminated by introducing limiters acting on gradients in space and time in a manner similar to that recently advocated (Sidilkover, 1998). Two schemes result from this analysis. The first is unconditionally TVD, but is second order accurate only under a CFL-like condition. The second scheme is second order accurate but subject to a CFL-like condition to maintain the TVD property. Results are presented for smooth and discontinuous solutions.