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

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Abstract
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 conservat... Abstract 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 conservat...

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