# Good non oscilliatory derivatives for an exsisting grid

I'm calculating the entropy production of a shockwave by utilizing the equations: $$$$\sigma = J'_q\frac{\partial}{\partial x}\left(\frac{1}{T}\right) +\frac{1}{T}\frac{4\eta}{3}\left(\frac{\partial v}{\partial x}\right)^2$$$$ and $$$$\sigma = \frac{\partial \rho_s}{\partial t} +\frac{\partial J_s}{\partial x}$$$$ I have access to all the variables on a spatial $$[x_i,...x_N]$$ and temporal grid $$[t_i,...t_K]$$. Currently I've used central discretization on both the spatial and temporal grid. The method works for the first equation, but for the second there seems to be oscillations before and behind the shock wave. Are there any better discretization methods I can use? Should I use a different method in the temporal dimension?

• Besides being non-oscillatory, do you have any other measures of quality that you care about? How did you get these solution variables (implicit, explicit, high- or low-order, fixed length or adaptive spacing), and do you want to stay in the same mathematical context? Same question for your spacial discretization with appropriate changes for space vs time. Oct 7, 2021 at 15:11
• Never mind, I retract most of my questions above. central-time, central-space. I think I'd still like to know what non-linear solver scheme you're using and why you don't just refine until the wiggles go away? Even if you just use an implicit method, the matrix can't be that big on a fine(ish) spacial grid. Oct 7, 2021 at 15:17
• I've been comparing results for Non-equillibrium molecular dynamic (NEMD) simulations and modelled the Euler equations which I've been comparing the results for (for details about the euler model you can see my question history). The NEMD spatial grid and temporal grid is pretty much set, but for the Euler equation I can refine the grids to a much greater degree. Is there any way to avoid the oscillations in the NEMD grid without refining the transiet grid? Oct 8, 2021 at 6:29