Error Bounds for Boundary-Stopped Positive Cubature Schemes for Hamilton--Jacobi--Bellman Equations

We derive error bounds for a boundary-stopped positive cubature scheme for degenerate parabolic Hamilton–Jacobi–Bellman equations with Dirichlet data on bounded domains. The scheme balances antipodal branches at their first boundary contacts and uses a control-dependent effective time. Its stopped-barrier remainder is $O(\tau \mathrm{\Delta }{t}^{\gamma \mathrm{/}2})$, and the total mass of interpolated branches is at most $2\tau \mathrm{/}\mathrm{\Delta }t$, where $\tau$ is the effective time. After division by the effective time, these estimates give a consistency bound that is uniform as $\tau \to 0$. Under a common strict boundary barrier and the stated mesh conditions, interior consistency, coefficient shaking, switching, and localized comparison yield $$\mathrm{\parallel }(u-u_h{)}^{+}{\mathrm{\parallel }}_{\mathrm{\infty }}\le C(\mathrm{\Delta }{t}^{1\mathrm{/}4}+h\mathrm{\Delta }{t}^{-1\mathrm{/}2}),\mathrm{\parallel }(u_h-u{)}^{+}{\mathrm{\parallel }}_{\mathrm{\infty }}\le C(\mathrm{\Delta }{t}^{1\mathrm{/}10}+h^{1\mathrm{/}2}\mathrm{\Delta }{t}^{-1\mathrm{/}4})\mathrm{.}$$ Thus $\mathrm{\Delta }t\asymp h^{10\mathrm{/}7}$ gives an $O(h^{1\mathrm{/}7})$ maximum-norm bound. The analysis applies to any fixed centrally symmetric positive degree-two cubature and permits rank-deficient diffusion. Numerical examples examine parallel heat loss to a cold wall and control-dependent, rank-deficient diffusion.