Boundary conditions

For walls and for the internal-flow inlet/outlet conditions Gyre sets $\mathbf U_L = \mathbf U_R = \mathbf U_\Gamma(\mathbf U^-)$. Any consistent numerical flux then returns exactly the physical flux of the boundary state, $\mathbf n\cdot\mathbf F(\mathbf U_\Gamma)$, with no dissipation. For a slip wall

so the wall flux is a pure pressure force with $p_\Gamma = p(\mathbf U_\Gamma)$. Following Hartmann and Leicht, a boundary flux of the form $\mathbf n\cdot\mathbf F(\mathbf U_\Gamma(\mathbf U^-))$, combined with functionals evaluated from the same boundary flux, yields an adjoint-consistent discretization: the discrete adjoint is a consistent discretization of the continuous adjoint problem including its boundary conditions. The mirrored-ghost-state alternative is available for comparison but is not adjoint consistent in this sense.

Forces are not recomputed from interpolated pressure. SurfaceOps.face_forces calls the residual kernel with faces=True and integrates the momentum components of the numerical boundary flux — inviscid minus viscous — that the residual itself used. Two consequences:

1. the force is discretely conservative: summing the surface loads and the far-field momentum flux reproduces the residual sum exactly;
2. the functional is adjoint consistent, a prerequisite for the superconvergence measured in the output-error study.

The boundary state keeps density and total energy and sets the velocity to zero, $\mathbf U_\Gamma = (\rho, \mathbf 0, \rho E)$ (adiabatic), or replaces the energy by $\rho\,c_v T_w$ (isothermal). Velocity (and wall temperature) enter the BR2 lifting as Dirichlet data, $\mathbf W^* = \mathbf W(\mathbf U_\Gamma)$; the viscous boundary flux is $\mathbf F_v(\mathbf U_\Gamma, \mathbf G + \eta\,\mathbf r_f)\cdot\mathbf n$, with the energy component removed for adiabatic walls (zero heat flux and zero velocity leave no energy transfer). The numerical energy flux through an adiabatic wall is therefore zero by construction, for any state. The cylinder validation checks instead that the wall heat conduction of the converged solution, the flux that is removed, decreases between the two highest polynomial degrees; it vanishes for the exact solution.

Boundary functions are ordinary JAX code of $(\mathbf U^-, \mathbf n, \mathbf x, \theta)$. Their dependence on flow parameters (the free-stream state in farfield, the back pressure in pressure_outlet) and on geometry (through $\mathbf n$) is therefore differentiated together with everything else. The linearization study verifies JVPs, VJPs and the block preconditioner for every wall type, the mirrored slip wall (slip_wall_mirror) with each interface flux included, for a channel mixing four different conditions and for an inlet/outlet channel (total_inlet, pressure_outlet, symmetry, slip_wall_mirror).