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.

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April 18, 2017, 06:34		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,882 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