CFL-monotone upwind update for first-order HJB equations

Use the model and coefficient notation of Periodic first-order controlled Hamilton–Jacobi Cauchy problem and the periodic grid of Periodic Cartesian grids and tensor-product multilinear interpolation. Choose an integer $M\ge 1$, set $\tau =T\mathrm{/}M$, and put $t_n=n\tau$ for $n=0,\dots ,M$. Choose a finite nonempty control set $A_k\subseteq A$. For a finite control model take $A_k=A$; for compact controls use a sequence of finite nonempty sets with the stated uniform Hamiltonian error. For each control and grid node put $b_{a,j}^{+}=\max \{b_{a,j},0\},b_{a,j}^{-}=\max \{-b_{a,j},0\},$ and define the explicit row $G_i^a(V)=V_i+\tau [{\sum }_{j=1}^d(\frac{b_{a,j}^{+}(x_i,t_n)}{h_j}(V_{i+e_j}-V_i)+\frac{b_{a,j}^{-}(x_i,t_n)}{h_j}(V_{i-e_j}-V_i))+{\mathrm{\ell }}_a(x_i,t_n)]\mathrm{.}$ The periodic HJB upwind update at time $t_n$ is $U_i^{n+1}={\min }_{a\in A_k}{G}_i^a(U^n)\mathrm{.}$ At every step $n=0,\dots ,M-1$, require the CFL condition $\tau {\max }_{a,i}{\sum }_{j=1}^d\frac{\mathrm{\mid }{b}_{a,j}(x_i,t_n)\mathrm{\mid }}{h_j}\le 1$ Initialize $U_i^0=u_0(x_i)$ and perform the updates for $n=0,\dots ,M-1$. Return $U^M$. Since the transport velocity in Periodic advection schemes and the finite-interval upwind row is $-b_a$ for this Hamiltonian sign convention, the displayed one-sided direction is its sign-reflected upwind row.