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We present an unconditionally energy-stable scheme for approximating theincompressible Navier-Stokes equations on domains with outflow/open boundaries.The scheme combines the generalized Positive Auxiliary Variable (gPAV) approachand a rotational velocity-correction type strategy, and the adoption of theauxiliary variable simplifies the numerical treatment for the open boundaryconditions. The discrete energy stability of the proposed scheme has beenproven, irrespective of the time step sizes. Within each time step the schemeentails the computation of two velocity fields and two pressure fields, bysolving an individual de-coupled Helmholtz (including Poisson) type equationwith a constant pre-computable coefficient matrix for each of these fieldvariables. The auxiliary variable, being a scalar number, is given by awell-defined explicit formula within a time step, which ensures the positivityof its computed values. Extensive numerical experiments with several flowsinvolving outflow/open boundaries in regimes where the backflow instabilitybecomes severe have been presented to test the performance of the proposedmethod and to demonstrate its stability at large time step sizes.

A gPAV-Based Unconditionally Energy-Stable Scheme for Incompressible Flows with Outflow/Open Boundaries