Explicit monotone diagonal-diffusion Bellman grid update

Use the model of Periodic diagonal controlled-diffusion HJB Cauchy problem and a periodic grid with integers $N_j\ge 3$, $h_j=1\mathrm{/}{N}_j$. Choose an integer $M\ge 1$, set $\tau =T\mathrm{/}M$, and put $t_n=n\tau$ for $n=0,\dots ,M$. For a nodal array $V$, define $D_{jj}^h{V}_i=\frac{V_{i+e_j}-2V_i+V_{i-e_j}}{h_j^2}\mathrm{.}$ For a control $a$, define $D_{j,a}^{up}{V}_i=\{\begin{array}{ll}(V_i-V_{i-e_j})\mathrm{/}{h}_j,& b_j^a(x_i,t_n)\ge 0,\\ (V_{i+e_j}-V_i)\mathrm{/}{h}_j,& b_j^a(x_i,t_n)<0.\end{array}$ The recurrence is $U_i^{n+1}=U_i^n+\tau {\max }_{a\in A}\{{\sum }_{j=1}^d[{\alpha }_j^a(x_i,t_n)D_{jj}^h{U}_i+b_j^a(x_i,t_n)D_{j,a}^{up}{U}_i]+f^a(x_i,t_n)\}\mathrm{.}$ At each time level $n=0,\dots ,M-1$, admit the update only when, at every node, control, and coordinate, $\mathrm{\mid }{b}_j^a(x_i,t_n)\mathrm{\mid }{h}_j\le {\alpha }_j^a(x_i,t_n),$ and $\tau {\max }_{i,a}{\sum }_{j=1}^d(\frac{2{\alpha }_j^a(x_i,t_n)}{h_j^2}+\frac{\mathrm{\mid }{b}_j^a(x_i,t_n)\mathrm{\mid }}{h_j})\le 1.$ Initialize by $U_i^0=u_0(x_i)$ and perform the updates for $n=0,\dots ,M-1$, evaluating coefficients at $(x_i,t_n)$. Return $U^M$. No claim about an arbitrary anisotropic central stencil is included.