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One beginner's question about discretization of a Hamilton-Jacobi equation(non-linear) $$ u_t = H(u_x) $$

$u_x$ is discreated with 2nd order ENO-FD

1st order: $D_1^{\pm}u = \pm [u_{x\pm1} - u_x ] / \Delta_x$

2nd order: $$D_2^{\pm}u = D_1^{\pm}u \mp \frac 1 2 \Delta x \{\min[ \max (D_1^{\pm}D_1^{\pm}u,0),\max(D_1^{-}D_1^{+}u,0)] +\\ \max[ \min (D_1^{\pm}D_1^{\pm}u,0),\min(D_1^{-}D_1^{+}u,0)] \} $$ and $n$th order flux $\hat D^n_x u=\max(|\max(D^{-,n}_xu,0)|,|\min(D^{+,n}_xu,0)|)$

while $u_t$ is discreated with 2nd Order ENO-Runge-Kutta $$ \delta_1 u = \Delta t H (\hat D_x u) \\ \delta_2 u= \frac {1} {2} (\delta_1u+ \Delta t H (\hat D_x (u + \delta_1 u)) \\ $$ It seems $x=u + \delta_1 u$ in flux $\hat D_x$ is not located at any grid point in stencil? And how to calculate $\hat D_x(u+\delta_1u)$?

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  • $\begingroup$ How are you discretizing $H(u_x)$ so that values are not returned on the same grid? Almost every finite difference method is constructed in this way. $\endgroup$
    – whpowell96
    Commented Dec 4, 2023 at 16:58
  • $\begingroup$ $D_x^{\pm} = \pm[{u_{x\pm1}-} u_x]/{\Delta x} $ and $\hat D_x u = \max(|\max(D_x^-u,0)|, |\min(D^+_xu, 0)| )$ spatial grid discretizing with same$\Delta x$, $[x_i, x_i+\Delta x]$. $\endgroup$
    – solanin
    Commented Dec 5, 2023 at 3:40
  • $\begingroup$ Where did you come up with $x=u+δ_1u$? On the left side you have the fixed grid coordinates, on the right a different function on the grid that evolves dynamically. It might help to once write all formulas with all arguments, starting with $u=u(x,t)$. // Is it intentional that there are no negative values in the ENO-FD, even if the function is falling? $\endgroup$ Commented Dec 5, 2023 at 10:12
  • $\begingroup$ That's right, I just need to calculate right hand side to update soultion. This item $u+\delta x$ confused me. $\endgroup$
    – solanin
    Commented Dec 5, 2023 at 14:54

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