PLM for Euler equations

sammy2 · April 17, 2017, 20:34

I'm trying to add second order reconstruction based on the following Taylor expansions for the edge states from some advection notes:

$a^{n+1/2}_{i+1/2,L} = a^n_i + \frac{\Delta x}{2} \frac{\partial a}{\partial x}\Big| _i + \frac{\Delta t}{2} \frac{\partial a}{\partial t}\Big| _i$

$a^{n+1/2}_{i+1/2,R} = a^n_{i+1} + \frac{\Delta x}{2} \frac{\partial a}{\partial x}\Big| _{i+1} + \frac{\Delta t}{2} \frac{\partial a}{\partial t}\Big| _{i+1}$

My plan is to apply these two equations to each of my conserved quantities (rho, rho*u, E) and use the spatial derivative of the corresponding quantity in the Euler equations (rho*u, rho*u*u+p, u(E+p)) for the time derivative.
Like so

$(\rho u)^{n+1/2}_{i+1/2,L} = (\rho u)^n_i + \frac{\Delta x}{2} \frac{\partial (\rho u)}{\partial x}\Big| _i + \frac{\Delta t}{2} \Big(-\frac{\partial(\rho u^2 + p)}{\partial x}\Big)\Big| _i$

Then use something like centered differences for each of the spatial derivatives before converting back to the primitive variables for the Riemann solver.
However, I've never seen it done like this so I wonder if this is wrong or there's a better way to implement piecewise linear reconstruction without using eigenvalues/characteristics?

Edit: I'm using finite volume discretization.

FMDenaro · April 18, 2017, 06:34

I am not sure if you are trying to do a Lax-Wendroff like discretization...

However:
1) for non linear equations it requires much more terms
2) Being based on the Taylor expansion, that cannot be applied in case of non regular solutions such as those that can arise in Euler equations
3) you need to write the integral form and define a unique flux function on each face

April 17, 2017, 20:34	PLM for Euler equations	#1
sammy2 New Member Join Date: Mar 2017 Posts: 7 Rep Power: 9	I'm trying to add second order reconstruction based on the following Taylor expansions for the edge states from some advection notes: $a^{n+1/2}_{i+1/2,L} = a^n_i + \frac{\Delta x}{2} \frac{\partial a}{\partial x}\Big\| _i + \frac{\Delta t}{2} \frac{\partial a}{\partial t}\Big\| _i$ $a^{n+1/2}_{i+1/2,R} = a^n_{i+1} + \frac{\Delta x}{2} \frac{\partial a}{\partial x}\Big\| _{i+1} + \frac{\Delta t}{2} \frac{\partial a}{\partial t}\Big\| _{i+1}$ My plan is to apply these two equations to each of my conserved quantities (rho, rhou, E) and use the spatial derivative of the corresponding quantity in the Euler equations (rhou, rhouu+p, u(E+p)) for the time derivative. Like so $(\rho u)^{n+1/2}_{i+1/2,L} = (\rho u)^n_i + \frac{\Delta x}{2} \frac{\partial (\rho u)}{\partial x}\Big\| _i + \frac{\Delta t}{2} \Big(-\frac{\partial(\rho u^2 + p)}{\partial x}\Big)\Big\| _i$ Then use something like centered differences for each of the spatial derivatives before converting back to the primitive variables for the Riemann solver. However, I've never seen it done like this so I wonder if this is wrong or there's a better way to implement piecewise linear reconstruction without using eigenvalues/characteristics? Edit: I'm using finite volume discretization.

Similar Threads
Thread	Thread Starter	Forum	Replies	Last Post
Guide: Writing Equations in LaTeX on the CFD Online Forums	pete	Site Help, Feedback & Discussions	27	May 19, 2022 04:19
SIMPLE Algorithm Finite Difference Equations: how to discretize and solve?	DA6righthand	Main CFD Forum	0	August 3, 2015 13:12
modelling Differential equations in a udf	RikardMNorén	Fluent UDF and Scheme Programming	2	October 1, 2013 04:36
Riemann invariants of adjoint equations of shallow water equations	zqb0929	Main CFD Forum	0	March 15, 2012 01:54
CFD governing equations	m.gos	Main CFD Forum	0	April 30, 2011 15:21

April 18, 2017, 06:34		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,849 Rep Power: 73	I am not sure if you are trying to do a Lax-Wendroff like discretization... However: 1) for non linear equations it requires much more terms 2) Being based on the Taylor expansion, that cannot be applied in case of non regular solutions such as those that can arise in Euler equations 3) you need to write the integral form and define a unique flux function on each face