research · note · in motion
What SBP operators are actually doing at a boundary
A field engineer's read of summation-by-parts: where the dual norm earns its name, and where it doesn't.
The dual norm, where it earns its name
SBP operators give you a discrete summation-by-parts identity. The catch: they only give you that identity in a specific norm. When the scheme is set up properly that norm is also the energy norm of the problem, and a SAT-style boundary penalty term then gives you discrete energy stability for free. When the scheme is not set up properly, the same operator silently forfeits its order at the boundary, and the convergence rate quietly drops.
What I'm actually trying to prove
The proof I'm writing is for a third-order generalised Gregory closure on a non-uniform grid. The interior is straightforward; the boundary closure is where the dual-norm bookkeeping goes wrong if you don't watch the penalty parameter. The current draft holds for a model problem (linear advection); the next pass extends to advection-diffusion with the diffusion term handled symmetrically.
Next
Tighten the dual-norm proof for the third-order Gregory closure. Then regenerate the convergence-rate plot for the model PDE on three meshes, and check the slope against the theoretical 3.