This paper describes a technique for achieving accurate numerical simulations of advective transport at large Courant numbers using large time steps. The scheme is called ULTIMATE DISCUS and it implements Leonard's universal flux limiter and QUICKEST algorithms within a semi-Lagrangian treatment of advection. This enables the scheme to achieve monotonic solutions, mass conservation and, most importantly, high accuracy without any limit on the time step (or Courant number).
The results of numerical experiments of advection over a fixed distance show that the accuracy of the method increases with increasing spatial resolution and generally increases (but in a non-trivial manner) with increasing Courant number. Accuracy is exact at all integer values of Courant number; for Courant numbers increasing between zero and one, accuracy improves rapidly and monotonically; for other integer-integer ranges of Courant number there is a minimum of accuracy close to the mid-range value. This behaviour is explained in terms of the known accuracy of the QUICKSET algorithm as a function of Courant number and the reducing number of interpolative steps required in the simulations as the Courant number increases. The use of the flux limiter is shown to remove non-physical oscillations from the solution, but at the price of a few per cent reduction in global accuracy caused by increased suppression of peak values.