A Second Order Accurate, Space-Time Limited, BDF Scheme
for the Linear Advection Equation
Shaun A Forth
Published in
Godunov Methods: Theory and Applications Ed. E.F. Toro
Kluwer Academic/Plenum Publishers 2001, p 335-352
Kluwer Academic/Plenum Publishers 2001, p 335-352
Presented at
International Conference on Godunov Methods: Theory
& Applications, Oxford 1999.
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 discretisation 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 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.
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 discretisation 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 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 variables in the time direction and it may
be eliminated by introducing limiters acting on gradients in space and time in
a manner simlar to that recently advocated (Sidilkover, 1998). Two
schemes result from this analysis. The first is unconditionally TVD, but
is second order accurate only unde 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.