A Second Order Accurate, Space-Time Limited, BDF Scheme for the Linear Advection Equation

Abstract

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 ad-vection. 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 vari 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.