How to compute WENO-reconstructed flux in local characteristic field?

OlegSutyrin · August 20, 2017, 04:02

I'm trying to apply finite-difference characteristic-wise WENO method to 1D Euler equations:
$U_t + F(U)_x = 0, \; \;\text{where} \;\; U = (\rho, \rho u, e) \;\; \text{and} \;\;F(U) = (\rho u, \rho u^2+p, (e+p)u)$
Physical values are set at grid points

, so we need to approximate fluxes at mid-points $x_{j\pm \frac{1}{2}}$ :
$U_t \approx \frac{1}{h} \left[ \widehat{f}_{j+\frac{1}{2}} - \widehat{f}_{j-\frac{1}{2}} \right]$
where $\widehat{f}_{j\pm\frac{1}{2}}$ are sought-for approximation of physical fluxes $f(U(x_{j\pm\frac{1}{2}}))$ .

My algorithm is the following:

1) For each $x_{j+\frac{1}{2}}$ , we calculate simple average state $U_{j+\frac{1}{2}} = \frac{1}{2} (U_j + U_{j+1})$ and, using it, local eigenvalues $(u-a, u, u+a)_{j+\frac{1}{2}}$ , right eigenvector matrix $R_{j+\frac{1}{2}}$ and it's left counterpart $R^{-1}_{j+\frac{1}{2}}$ . (Subscript ${j+\frac{1}{2}}$ will be omitted below)

2) Now we transform

, its differences $\Delta U$ and flux differences $\Delta F(U)$ to local characteristic field:
$W=R^{-1}U, \;\; \Delta W=R^{-1}\Delta U, \;\; \Delta F(W)=R^{-1}\Delta F(U)$
It is done only for relevant grid points which in my case (

) are $x_{j-1}, x_j, x_{j+1}, x_{j+2}$ .

3) Then we reconstruct characteristic variable values by WENO method using

, $\Delta W$ and $\Delta F(W)$ . We get two "candidates":
$\widehat{W}^- = R_j(x_{j+\frac{1}{2}}) \; \text{and} \; \widehat{W}^+ = R_{j+1}(x_{j+\frac{1}{2}})$
where $R_j(x), R_{j+1}(x)$ are a convex combinations of corresponding polynomial functions obtained by WENO method.

The next step would be to compute 2-point flux function $h(\widehat{W}^-,\widehat{W}^+)$ using these reconstructed values. I'm using Lax-Friedrich's function:
$h(a,b) = \frac{1}{2} \left[ F(a) + F(b) + \alpha(b-a) \right]$
where $\alpha$ is maximum value of eigenvalue (different for each of equations, since they are decoupled now).

But how to compute these

and

using $\widehat{W}^\pm$ ? In component-wise approach it is simple: we reconstruct base values - $(\widehat{\rho}^\pm, \widehat{u}^\pm, \widehat{p}^\pm)$ , for example - and the use explicit formulas to compute $F(a) = (\widehat{\rho}^- \widehat{u}^-, \; \widehat{\rho}^- (\widehat{u}^-)^2 + \widehat{p}^-,\; ...)$ and $F(b) = (\widehat{\rho}^+ \widehat{u}^+, \; \widehat{\rho}^+ (\widehat{u}^+)^2 + \widehat{p}^+,\; ...)$ , but we don't have such formulas for $F(\widehat{W})$ because $\widehat{W}$ is expressed in local characteristic field...

OlegSutyrin · August 22, 2017, 06:03

It seems that I was misunderstanding how the flux is reconstructed:

My idea was to reconstruct base values -

(or

if we are performing characteristic decomposition) - and then substitute them to formulas of

(or

) to obtain reconstructed fluxes.

It looks like the correct way is to apply reconstruction procedure to

(or

which is obtained by transformation $F(W) = R^{-1}F(U)$ ) itself. Some quick tests I performed with this approach show very plausible results.

usv001 · September 4, 2017, 14:24

Hi there,

Here's what I am doing: after reconstructing the characteristic variables at the face, I use the left eigenvector matrix to convert them back to primitive variables. Then, I use these to compute the flux using the flux function, e.g. Lax-Friedrich method. It has performed well in several 1D test cases.

USV

August 20, 2017, 04:02	How to compute WENO-reconstructed flux in local characteristic field?	#1
OlegSutyrin Member Oleg Sutyrin Join Date: Feb 2016 Location: Russia Posts: 41 Rep Power: 10	I'm trying to apply finite-difference characteristic-wise WENO method to 1D Euler equations: $U_t + F(U)_x = 0, \; \;\text{where} \;\; U = (\rho, \rho u, e) \;\; \text{and} \;\;F(U) = (\rho u, \rho u^2+p, (e+p)u)$ Physical values are set at grid points , so we need to approximate fluxes at mid-points $x_{j\pm \frac{1}{2}}$ : $U_t \approx \frac{1}{h} \left[ \widehat{f}_{j+\frac{1}{2}} - \widehat{f}_{j-\frac{1}{2}} \right]$ where $\widehat{f}_{j\pm\frac{1}{2}}$ are sought-for approximation of physical fluxes $f(U(x_{j\pm\frac{1}{2}}))$ . My algorithm is the following: 1) For each $x_{j+\frac{1}{2}}$ , we calculate simple average state $U_{j+\frac{1}{2}} = \frac{1}{2} (U_j + U_{j+1})$ and, using it, local eigenvalues $(u-a, u, u+a)_{j+\frac{1}{2}}$ , right eigenvector matrix $R_{j+\frac{1}{2}}$ and it's left counterpart $R^{-1}_{j+\frac{1}{2}}$ . (Subscript ${j+\frac{1}{2}}$ will be omitted below) 2) Now we transform , its differences $\Delta U$ and flux differences $\Delta F(U)$ to local characteristic field: $W=R^{-1}U, \;\; \Delta W=R^{-1}\Delta U, \;\; \Delta F(W)=R^{-1}\Delta F(U)$ It is done only for relevant grid points which in my case () are $x_{j-1}, x_j, x_{j+1}, x_{j+2}$ . 3) Then we reconstruct characteristic variable values by WENO method using , $\Delta W$ and $\Delta F(W)$ . We get two "candidates": $\widehat{W}^- = R_j(x_{j+\frac{1}{2}}) \; \text{and} \; \widehat{W}^+ = R_{j+1}(x_{j+\frac{1}{2}})$ where $R_j(x), R_{j+1}(x)$ are a convex combinations of corresponding polynomial functions obtained by WENO method. The next step would be to compute 2-point flux function $h(\widehat{W}^-,\widehat{W}^+)$ using these reconstructed values. I'm using Lax-Friedrich's function: $h(a,b) = \frac{1}{2} \left[ F(a) + F(b) + \alpha(b-a) \right]$ where $\alpha$ is maximum value of eigenvalue (different for each of equations, since they are decoupled now). But how to compute these and using $\widehat{W}^\pm$ ? In component-wise approach it is simple: we reconstruct base values - $(\widehat{\rho}^\pm, \widehat{u}^\pm, \widehat{p}^\pm)$ , for example - and the use explicit formulas to compute $F(a) = (\widehat{\rho}^- \widehat{u}^-, \; \widehat{\rho}^- (\widehat{u}^-)^2 + \widehat{p}^-,\; ...)$ and $F(b) = (\widehat{\rho}^+ \widehat{u}^+, \; \widehat{\rho}^+ (\widehat{u}^+)^2 + \widehat{p}^+,\; ...)$ , but we don't have such formulas for $F(\widehat{W})$ because $\widehat{W}$ is expressed in local characteristic field...

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August 22, 2017, 06:03		#2
OlegSutyrin Member Oleg Sutyrin Join Date: Feb 2016 Location: Russia Posts: 41 Rep Power: 10	It seems that I was misunderstanding how the flux is reconstructed: My idea was to reconstruct base values - (or if we are performing characteristic decomposition) - and then substitute them to formulas of (or ) to obtain reconstructed fluxes. It looks like the correct way is to apply reconstruction procedure to (or which is obtained by transformation $F(W) = R^{-1}F(U)$ ) itself. Some quick tests I performed with this approach show very plausible results.

September 4, 2017, 14:24		#3
usv001 Senior Member Join Date: Sep 2015 Location: Singapore Posts: 102 Rep Power: 11	Hi there, Here's what I am doing: after reconstructing the characteristic variables at the face, I use the left eigenvector matrix to convert them back to primitive variables. Then, I use these to compute the flux using the flux function, e.g. Lax-Friedrich method. It has performed well in several 1D test cases. USV