Conservative Explicit Unrestricted-Time-Step Multidimensional Constancy-Preserving Advection Schemes

The multidimensional advection schemes described in this study are based on a strictly conservative flux-based control-volume formulation. They use an explicit forward-in-time update over a single time step, but there are no “stability” restrictions on the time step. Genuinely multidimensional forward-in-time advection schemes require an estimate of transverse contributions to each face-normal flux for stability. Traditional operator-splitting techniques based on sequential one-dimensional updates introduce such transverse cross-coupling automatically; however, they have serious shortcomings. For example, conservative-form operator splitting is indeed globally conservative but introduces a serious “splitting error”; in particular, a constant is not preserved in general solenoidal velocity fields. By contrast, advective-form operator splitting is constancy preserving but not conservative. However, by using advective-form estimates for the transverse contributions together with an overall conservative-form update, strictly conservative constancy-preserving schemes can be constructed. The new methods have the unrestricted-time-step advantages of semi-Lagrangian schemes, but with the important additional attribute of strict conservation due to their flux-based formulation. Shape-preserving techniques developed for small time steps can be incorporated. For large time steps, results are not strictly shape preserving but, in practice, deviations appear to be very slight so that overall behavior is essentially shape preserving. Since only one-dimensional flux calculations are required at each step of the computation, the algorithms described here should be highly compatible with existing advection codes based on conventional operator-splitting methods. Capabilities of the new schemes are demonstrated using the well-known scalar advection test problem devised by Smolarkiewicz.