An eikonal kink and its monotone HJB grid updates

Away from $x=0,1\mathrm{/}2$ and the moving fronts $d_{{\mathbb{T}}^1}(x,0)=t$, the function is either constant zero or has $u_t=-1$ and $\mathrm{\mid }{u}_x\mathrm{\mid }=1$; it satisfies the PDE classically. Near a moving front on the increasing-distance side, put $s=x-t$ in a local lift; on the other side put $s=1-x-t$. In either case $u=\max (s,0)$. There is no $C^1$ upper test at this convex kink: its directional slopes would have to be at least $1$ in the positive $s$ direction and at most $0$ in the negative direction. Any $C^1$ lower test has $s$-slope $p\in [0,1]$. Use a second coordinate tangent to the moving front: the profile is constant in that direction, so the test's tangential derivative is zero. Thus ${\varphi }_t=-p$ and ${\varphi }_x=p$ on the increasing-distance side, or ${\varphi }_x=-p$ on the decreasing-distance side; in both cases its residual is $-p+\mathrm{\mid }p\mathrm{\mid }=0$. At the antipode $x=1\mathrm{/}2$ for $t<1\mathrm{/}4$, the local profile is $1\mathrm{/}2-t-\mathrm{\mid }x-1\mathrm{/}2\mathrm{\mid }$. Its upper tests have spatial slope $p\in [-1,1]$ and time derivative $-1$, giving residual $-1+\mathrm{\mid }p\mathrm{\mid }\le 0$; no lower test exists at that concave cusp. Thus the sub- and supersolution tests hold everywhere for $0<t<1\mathrm{/}4$. The initial trace is $u_0$.

At $x=0$, the sampled data have $U_0=0$ and $U_{\pm 1}=h$ because $u_0(\pm h)=h$. For either semi-Lagrangian action the footpoint is $\pm \tau$; linear interpolation between $0$ and $\pm h$ returns $\tau$ when $\tau \le h$, so the action minimum is $\tau$. For the upwind update the forward difference is $1$ and the backward difference is $-1$; multiplying by $a=1$ and $a=-1$, respectively, gives drift contribution $1$ for both actions. The CFL condition is $\tau \mathrm{/}h\le 1$, so its minimum update is also $\tau$. Since $u(0,\tau )=0$, both one-step errors equal $\tau$. Taking $\tau =h\mathrm{/}2$ gives $h^2\mathrm{/}\tau =2h\to 0$, and [F3] supplies convergence for the refined semi-Lagrangian family under its full hypotheses.