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

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Old   August 20, 2017, 04:02
Default How to compute WENO-reconstructed flux in local characteristic field?
Oleg Sutyrin
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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 x_j, 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 U, 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 (r=2) are x_{j-1}, x_j, x_{j+1}, x_{j+2}.

3) Then we reconstruct characteristic variable values by WENO method using W, \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 F(a) and F(b) 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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Old   August 22, 2017, 06:03
Oleg Sutyrin
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It seems that I was misunderstanding how the flux is reconstructed:

My idea was to reconstruct base values - U (or W if we are performing characteristic decomposition) - and then substitute them to formulas of F(U) (or F(W)) to obtain reconstructed fluxes.

It looks like the correct way is to apply reconstruction procedure to F(U) (or F(W) which is obtained by transformation F(W) = R^{-1}F(U)) itself. Some quick tests I performed with this approach show very plausible results.
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Old   September 4, 2017, 14:24
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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.

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charateristic-wise, finite-difference, weno

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