Abstract
Reduced basis methods (RB methods or RBMs) form one of the most promising
techniques to deliver numerical solutions of parametrized PDEs in real-time with reasonable
accuracy [1]. For the Navier-Stokes equation, RBMs based on stable velocity-pressure spaces do
not generally inherit the stability of the high-fidelity method. Common techniques for working
around this problem (e.g. [2]) have the effect of deteriorating the performance of the RBM in
the performance-critical online stage.
We show how divergence-free reduced formulations eliminates this problem, producing
RBMs that are faster by an order of magnitude or more in the online stage. This is most
easily achieved using divergence-conforming compatible B-spline bases, using a transformation
that can maintain the divergence-free property under variable geometries. See [3] for more
details.
We also demonstrate the flexibility of RBMs for non-stationary flow problems using a
problem with two stages: an initial, finite transient stage where the flow pattern settles from
the initial data, followed by a terminal and infinite oscillatory stage characterized by vortex
shedding. We show how an RBM whose data is only sourced from the terminal stage nevertheless
can produce solutions that pass through the initial stage without critical problems (e.g. crashing,
diverging or blowing up).
techniques to deliver numerical solutions of parametrized PDEs in real-time with reasonable
accuracy [1]. For the Navier-Stokes equation, RBMs based on stable velocity-pressure spaces do
not generally inherit the stability of the high-fidelity method. Common techniques for working
around this problem (e.g. [2]) have the effect of deteriorating the performance of the RBM in
the performance-critical online stage.
We show how divergence-free reduced formulations eliminates this problem, producing
RBMs that are faster by an order of magnitude or more in the online stage. This is most
easily achieved using divergence-conforming compatible B-spline bases, using a transformation
that can maintain the divergence-free property under variable geometries. See [3] for more
details.
We also demonstrate the flexibility of RBMs for non-stationary flow problems using a
problem with two stages: an initial, finite transient stage where the flow pattern settles from
the initial data, followed by a terminal and infinite oscillatory stage characterized by vortex
shedding. We show how an RBM whose data is only sourced from the terminal stage nevertheless
can produce solutions that pass through the initial stage without critical problems (e.g. crashing,
diverging or blowing up).