I am considering solving HJB equation of the form $$ v_t=g(a(x)v_x),\quad x\in \mathbb{R}, t>0, $$ with initial condition $v(0)=v_0$. Here $g:\mathbb{R}\to \mathbb{R}$ is Lipschitz and monotone increasing. I hope to construct a scheme stable in $L^\infty$ norm.

Here is what I tried. For the linear $g$, i.e. $g(p)=cp$ with possible space dependent $c>0$, I consider the upwind and downwinding scheme; $$ D_iv^{n}=\begin{cases}\frac{v_{i+1}^{n}-v_{i}^{n}}{h},& a_i>0,\\ \frac{v_{i}^{n}-v_{i-1}^{n}}{h},& a_i<0 \end{cases}, $$ and update the solution by $$v^{n+1}_i=v^{n}_i+\tau c_ia_iD_iv^n.$$ Then for $a_i>0$, we have $$v^{n+1}_i=v^{n}_i+\frac{\tau c_ia_i}{h}(v_{i+1}^{n}-v_{i}^{n})=(1-C_i)v_i^n+C_iv^{n}_{i+1}\le |v^n|_\infty,$$ with $C_i=\frac{\tau c_ia_i}{h}>0$, while for $a_i<0$, we have $$v^{n+1}_i=v^{n}_i+\frac{\tau c_ia_i}{h}(v_{i}^{n}-v_{i-1}^{n})=(1-C_i)v_i^n+C_iv^{n}_{i+1}\le |v^n|_\infty,$$ with $C_i=-\frac{\tau c_ia_i}{h}>0$.

However, for Lipschitz and monotone increasing $g$, the scheme becomes $$v^{n+1}_i=v^{n}_i+\tau g(a_iD_iv^n).$$ To perform similar analysis, I write $$g(a_iD_iv^n)=g(a_iD_iv^n)-g(0)+g(0),$$ thus $$|v_i^{n+1}|=|v_i^n+\tau (g(a_iD_iv^n)-g(0))|+\tau|g(0)|,$$ then using Lipschitz $g$ I have $$|v_i^{n+1}|=|v_i^n|+L\tau |a_iD_iv^n|+\tau|g(0)|.$$

But the similar analysis can't process since I don't know how to write $|v^{n}_i|+C_i|v^{n}_{i+1}-v^{n}_{i}|$ as a convex combination. Also I haven't used the monotone increasing assumption.

I have also tried the scheme discussed here, but I still can't demonstrate the stability due to a similar reason.

Is my simple upwinding scheme a stable scheme to solve this problem? I am not sure whether the proof fails due to my poor techniques or the scheme is not stable in general for this type of $g$. Is there any other scheme commonly used for this problem?


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