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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.