Convergence of the positive-interpolation and upwind first-order HJB schemes

By the stability part of [F1] and $M_k{\tau }_k=T$, all nodal values satisfy $\mathrm{\parallel }{U}_k^n{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{\mathrm{\infty }}+L{t}_n\le \mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{\mathrm{\infty }}+LT\mathrm{.}$ The convex interpolation weights give the same bound for $V_k$, so the family is locally bounded on ${\mathbb{T}}^d\times [0,T)$. Form its upper and lower half-relaxed limits $\bar{V},\underset{‾}{V}$ as in [F2].

Let $\bar{V}-\varphi$ have a local maximum at an interior point $(x_0,t_0)$, with $\varphi$ smooth. Add $\epsilon (\mathrm{\mid }x-x_0{\mathrm{\mid }}^4+\mathrm{\mid }t-t_0{\mathrm{\mid }}^4)$ to $\varphi$; this makes the contact strict without changing its first derivatives at $(x_0,t_0)$, as in [F4]. Choose the nodal local maxima $(x_k,t_k)$ in [F3], with $(x_k,t_k)\to (x_0,t_0)$; write $c_k=U_{i_k}^{n_k}-\varphi (x_k,t_k)$. For all sufficiently large $k$, every node used by the previous-time update at $(x_k,t_k)$ lies inside the fixed contact ball: the upwind neighbors are at distance at most $h_{k,\max }$ and each semi-Lagrangian interpolation vertex is at distance at most ${\tau }_{k}B+\sqrt{d}{h}_{k,\max }$. Hence each such previous-layer nodal value satisfies $U_j^{n_k-1}\le \varphi (x_j,t_k-{\tau }_k)+c_k$. Order preservation and constant-shift invariance from [F1] imply $U_{i_k}^{n_k}=S_k(U^{n_k-1}{)}_{i_k}\le S_k(\varphi (\cdot ,t_k-{\tau }_k)+c_k{)}_{i_k}=S_k(\varphi (\cdot ,t_k-{\tau }_k){)}_{i_k}+c_k\mathrm{.}$ Since $U_{i_k}^{n_k}=\varphi (x_k,t_k)+c_k$, the normalized residual is nonpositive. Taking $k\to \mathrm{\infty }$ and using the consistency estimate in [F1] gives ${\varphi }_t(x_0,t_0)+H(x_0,t_0,D\varphi (x_0,t_0))\le 0$. By [F4], $\bar{V}$ is a viscosity subsolution for the full $C^1$ test class.

At a local minimum of $\underset{‾}{V}-\varphi$, subtract $\epsilon (\mathrm{\mid }x-x_0{\mathrm{\mid }}^4+\mathrm{\mid }t-t_0{\mathrm{\mid }}^4)$ from $\varphi$ to make the contact strict without changing its derivatives at $(x_0,t_0)$, as in [F4]. Use the minimizing nodes in [F3]. All previous-layer stencil nodes lie in the contact ball, so $U_j^{n_k-1}\ge \varphi (x_j,t_k-{\tau }_k)+c_k$. Order preservation and constant-shift invariance now yield the reverse update inequality, hence the normalized consistency residual is nonnegative. Passing to the limit gives the viscosity supersolution inequality. The smooth-test equivalence [F4] extends it to every $C^1$ lower test.

The initial trace estimate [F6] and the interpolation error [F7] show that both half-relaxed limits equal $u_0$ on $t=0$. They are bounded by step 1.1 and have the required semicontinuity by [F2]. Apply comparison [F5] to the periodic lifts of $\bar{V}$ and $\underset{‾}{V}$: it gives $\bar{V}\le \underset{‾}{V}$. The reverse inequality is part of [F2], so they coincide with a continuous viscosity solution having initial data $u_0$.

Fix $T^{\mathrm{\prime }}<T$ and let $K=[0,1{]}^d\times [0,T^{\mathrm{\prime }}]$ in the periodic lift; it is compact by [F8]. Fix $\epsilon >0$ and write $u$ for the common continuous limit from step 2.3. For $z\in K$ and $q\in {\mathbb{Q}}_{>0}$, put $P(z,q)$ when the set-defined upper and lower envelopes in [F2] give $\mathrm{\mid }{V}_k(y)-u(z)\mathrm{\mid }\le \epsilon \mathrm{/}3\text{for every integer }k\text{ with }1\mathrm{/}k<q\text{ and every }y\text{ with }\mathrm{\mid }y-z\mathrm{\mid }<q,$ and continuity gives $\mathrm{\mid }u(y)-u(z)\mathrm{\mid }<\epsilon \mathrm{/}3$ whenever $\mathrm{\mid }y-z\mathrm{\mid }<q$. The family of all balls $\{B(z,q\mathrm{/}2):z\in K,q\in {\mathbb{Q}}_{>0},P(z,q)\}$ covers $K$: each point has a positive real radius from the two envelope definitions and continuity, and a smaller rational radius still works. This is the family of all qualifying pairs, so no radius is selected for every point. By [F8] take a finite subcover and let $q_{\ast }>0$ be its least radius. For $1\mathrm{/}k<q_{\ast }$, every $y\in K$ lies in one of these balls and therefore $\mathrm{\mid }{V}_k(y)-u(y)\mathrm{\mid }<2\epsilon \mathrm{/}3<\epsilon$. This proves uniform convergence on the fundamental box through $T^{\mathrm{\prime }}$, hence on the torus and every compact subset of ${\mathbb{T}}^d\times [0,T)$.