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Hi I want to approximate $u(x,t)$ solution of:

$\begin{cases}u_t+H(u_x)=0 \\u(x,0)=u_0(x)= e^{−(x−3)^2} \end{cases}$, where

$H(u_x)=\frac{u_x^2}{2}$ and my spatial domain is $\Omega=[0;10]$.

Now i want to apply Hopf-Lax formula (Evans' chapter 3 theorem 4), who says

$u(x_j,t_n+\Delta t)=\min_{a\in \mathbb{R}} \left\{ \Delta t H^*(a)+u(x_j-a\Delta t,t_n)\right\} $

in my case $H^*(p)=H(p)=\frac{p^2}{2}$ and $u(x_j-a\Delta t,t_n)$ needs to be interpolated but... which different values I can assign to $a$?

I tested it with $a=\left\{ \min_j u(x_j,t_n),\max_j u(x_j,t_n)\right\}$ but it seems only partially working.

Any help?

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