On the Roe approximate Riemann solver in the preconditioned density based method

Eifoehn4 · March 14, 2022, 14:56

I don't know if this will help you:

Initially consider only the conservative Jacobian without preconditioner with

$\boldsymbol{Q}= \begin{pmatrix} \rho \\[0.2em] \rho u \\[0.2em] \rho e \end{pmatrix}= \begin{pmatrix} \alpha \\[0.2em] \beta \\[0.2em] \gamma \end{pmatrix}, \quad \boldsymbol{F}(\boldsymbol{Q})= \begin{pmatrix} \rho u \\[0.2em] \rho u^2 + p \\[0.2em] u \left( E + p \right) \end{pmatrix}= \begin{pmatrix} \beta \\[0.2em] \beta^2/\alpha + p \\[0.2em] \beta \gamma/\alpha + p \beta/\alpha \end{pmatrix},$

and the pressure $p=f(\alpha,\beta,\gamma)$ defined as a function of all conservative variables.

For non-ideal gases the crucial part is derive consistent Roe partial derivatives to fulfill following relation

$\text{d}p=\left. \frac{\partial{p}}{\partial{\alpha}} \right|_{\beta,\gamma} \text{d}\alpha + \left. \frac{\partial{p}}{\partial{\beta}} \right|_{\alpha,\gamma} \text{d}\beta + \left. \frac{\partial{p}}{\partial{\gamma}} \right|_{\alpha,\beta} \text{d}\gamma.$

One approach may be the Liou-Glaister generalization.

Here you initially calculate the partial derivatives point-wise with the EOS with the classical Roe averages (denoted with a tilde symbol).

They do not fulfill the first relation and result in a wrong $\Delta p$ (origin of instability).

$\Delta p \neq \tilde{p}_{\alpha}\Delta\alpha + \tilde{p}_{\beta}\Delta\beta + \tilde{p}_{\gamma}\Delta\gamma,$

After that you correct them by a projection method (denoted with a hat symbol)

$\partial p=- \Delta p + \tilde{p}_{\alpha}\Delta\alpha + \tilde{p}_{\beta}\Delta\beta + \tilde{p}_{\gamma}\Delta\gamma,$

where

$\hat{p}_{\alpha} = \tilde{p}_{\alpha} \left(1 + \tilde{p}_{\alpha} \Delta \alpha \partial p / D \right) ,$

$\hat{p}_{\beta} = ...$

$\hat{p}_{\gamma} = ...$

with

$D = (\tilde{p}_{\alpha}\Delta\alpha)^2+(\tilde{p}_{\beta}\Delta\beta)^2+(\tilde{p}_{\gamma}\Delta\gamma)^2.$

Now in analogy to that you may approach in a similar way for the preconditioner version by considering the two equations

$\text{d}\rho=\left. \frac{\partial{\rho}}{\partial{p}} \right|_{T} \text{d}p + \left. \frac{\partial{\rho}}{\partial{T}} \right|_{p} \text{d}T + ...$

and

$\text{d}H=\left. \frac{\partial{H}}{\partial{p}} \right|_{T} \text{d}p + \left. \frac{\partial{H}}{\partial{T}} \right|_{p} \text{d}T + ...$

to derive consistent primitive partial derivatives.

The approach is well described in the paper of Mottura.

Regards

sbaffini · March 15, 2022, 08:11

Quote:

Originally Posted by Eifoehn4

I don't know if this will help you:

Initially consider only the conservative Jacobian without preconditioner with

$\boldsymbol{Q}= \begin{pmatrix} \rho \\[0.2em] \rho u \\[0.2em] \rho e \end{pmatrix}= \begin{pmatrix} \alpha \\[0.2em] \beta \\[0.2em] \gamma \end{pmatrix}, \quad \boldsymbol{F}(\boldsymbol{Q})= \begin{pmatrix} \rho u \\[0.2em] \rho u^2 + p \\[0.2em] u \left( E + p \right) \end{pmatrix}= \begin{pmatrix} \beta \\[0.2em] \beta^2/\alpha + p \\[0.2em] \beta \gamma/\alpha + p \beta/\alpha \end{pmatrix},$

and the pressure $p=f(\alpha,\beta,\gamma)$ defined as a function of all conservative variables.

For non-ideal gases the crucial part is derive consistent Roe partial derivatives to fulfill following relation

$\text{d}p=\left. \frac{\partial{p}}{\partial{\alpha}} \right|_{\beta,\gamma} \text{d}\alpha + \left. \frac{\partial{p}}{\partial{\beta}} \right|_{\alpha,\gamma} \text{d}\beta + \left. \frac{\partial{p}}{\partial{\gamma}} \right|_{\alpha,\beta} \text{d}\gamma.$

One approach may be the Liou-Glaister generalization.

Here you initially calculate the partial derivatives point-wise with the EOS with the classical Roe averages (denoted with a tilde symbol).

$\Delta p = \tilde{p}_{\alpha}\Delta\alpha + \tilde{p}_{\beta}\Delta\beta + \tilde{p}_{\gamma}\Delta\gamma,$

They do not fulfill the first relation and result in a wrong $\Delta p$ (origin of instability). After that you correct them by a projection method (denoted with a hat symbol)

$\hat{p}_{\alpha} = \tilde{p}_{\alpha} \left(1 + \tilde{p}_{\alpha} \Delta \alpha \Delta p / D \right) ,$

$\hat{p}_{\beta} = ...$

$\hat{p}_{\gamma} = ...$

with

$D = (\tilde{p}_{\alpha}\Delta\alpha)^2+(\tilde{p}_{\beta}\Delta\beta)^2+(\tilde{p}_{\gamma}\Delta\gamma)^2.$

Now in analogy to that you may approach in a similar way for the preconditioner version by considering the two equations

$\text{d}\rho=\left. \frac{\partial{\rho}}{\partial{p}} \right|_{T} \text{d}p + \left. \frac{\partial{\rho}}{\partial{T}} \right|_{p} \text{d}T + ...$

and

$\text{d}H=\left. \frac{\partial{H}}{\partial{p}} \right|_{T} \text{d}p + \left. \frac{\partial{H}}{\partial{T}} \right|_{p} \text{d}T + ...$

to derive consistent primitive partial derivatives.

The approach is well described in the paper of Mottura.

Regards

I think I finally get how this should work. Still, I have a doubt about the fact that I would be applying a method intended to work for cases where the $\delta X^*$ at numerator over D is necessarily not 0, to a case where, instead, it should be (but isn't, maybe only because I don't know how to write it with my variables). However, I might still use the idea to actually check what I'm doing.

For now, my euristic explanation for the matter is that, looking at my final dissipation vector here, my d1 component, which appears on all the equations, starts with drho/dT and all the other terms have rho_ROE inside, so drho/dT should sort of have it as well.

sbaffini · March 19, 2022, 12:20

Quote:

Originally Posted by sbaffini

Just for your interest, this is what I have so far in my scheme for the dissipation part $\mathbf{d}=\mathbf{\Gamma}\mathbf{R}|\mathbf{\Lambda}|\mathbf{R}^{-1}(\mathbf{q}_R-\mathbf{q}_L)$ (but note that I don't use it in this form, I still use matrix products to avoid loosing control):

$d_1 = \rho_T|V_n|a+\rho b$
$d_2 = u d_1 + \rho|V_n|\left(u_R-u_L\right) + \rho n_x c$
$d_3 = v d_1 + \rho|V_n|\left(v_R-v_L\right) + \rho n_y c$
$d_4 = w d_1 + \rho|V_n|\left(w_R-w_L\right) + \rho n_z c$
$d_5 = H d_1 + \rho|V_n|e+ \rho V_n c$

with:

$a = \left(T_R-T_L\right)-\frac{\delta}{\rho C_p} \left(P_R-P_L\right)$
$b = \frac{\theta_2}{2\rho s_2V_r^2}\left(P_R-P_L\right)+\frac{\theta_3}{2s_2V_r^2}\left[\mathbf{n} \cdot \left(\mathbf{V}_R-\mathbf{V}_L\right)\right]$
$c = \frac{\theta_1}{2 \rho s_2} \left(P_R-P_L\right) +\left(\frac{\theta_2}{2 s_2} -|V_n| \right) \left[\mathbf{n} \cdot \left(\mathbf{V}_R-\mathbf{V}_L\right)\right]$
$e = C_p a + \mathbf{V} \cdot \left(\mathbf{V}_R-\mathbf{V}_L\right)$

and:
$\theta_0 = |s_1+s_2|+|s_1-s_2| = 2MAX\left(s1,s2\right)$
$\theta_1 = |s_1+s_2|-|s_1-s_2| = 2MIN\left(s1,s2\right)$
$\frac{\theta_2}{2s2} = \frac{\theta_1}{2s2}\left(V_n-s_1\right)-\frac{\theta_0}{2}$
$\frac{\theta_3}{2s2} = \frac{\theta_1}{2s2}\left[\left(V_n-s_1\right)^2+s_2^2\right]-\left(V_n-s_1\right)\theta_0$
$s_1 = \left(1-\alpha\right)V_n$
$s_2 = \sqrt{\left(\alpha V_n \right)^2+V_r^2}$
$\alpha = \frac{1-\beta V_r^2}{2}$

where $\beta=\rho_P+\rho T \delta /(\rho Cp)$ is the inverse of the speed of sound squared, $\delta = 1 - \rho H_p=-\rho_T T/\rho$ and anything without the R/L subscript to be evaluated at the Roe average. One could actually further simplify things by nothing that:

$V_n-s_1 = \alpha V_n$
$\left(V_n-s_1\right)^2+s_2^2=2 \alpha^2 V_n^2 + V_r^2$

but I've found difficulties in proceeding further toward a version which could even remotely resemble the Roe one (not sure if it is also due to the certainly different eigenvectors I'm using).

Just a note to mention that the dissipation vector I reported above was actually wrong (as I inverted the 4th and 5th eigenvectors in doing the products). However, the error just affects the coefficients appearing in b and c, basically just $\theta_2$ and $\theta_3$ . So, in the end, the heuristic reasoning on $\rho_T$ still looks legit.

March 14, 2022, 14:56		#21
Eifoehn4 Senior Member - Join Date: Jul 2012 Location: Germany Posts: 184 Rep Power: 14	I don't know if this will help you: Initially consider only the conservative Jacobian without preconditioner with $\boldsymbol{Q}= \begin{pmatrix} \rho \\[0.2em] \rho u \\[0.2em] \rho e \end{pmatrix}= \begin{pmatrix} \alpha \\[0.2em] \beta \\[0.2em] \gamma \end{pmatrix}, \quad \boldsymbol{F}(\boldsymbol{Q})= \begin{pmatrix} \rho u \\[0.2em] \rho u^2 + p \\[0.2em] u \left( E + p \right) \end{pmatrix}= \begin{pmatrix} \beta \\[0.2em] \beta^2/\alpha + p \\[0.2em] \beta \gamma/\alpha + p \beta/\alpha \end{pmatrix},$ and the pressure $p=f(\alpha,\beta,\gamma)$ defined as a function of all conservative variables. For non-ideal gases the crucial part is derive consistent Roe partial derivatives to fulfill following relation $\text{d}p=\left. \frac{\partial{p}}{\partial{\alpha}} \right\|_{\beta,\gamma} \text{d}\alpha + \left. \frac{\partial{p}}{\partial{\beta}} \right\|_{\alpha,\gamma} \text{d}\beta + \left. \frac{\partial{p}}{\partial{\gamma}} \right\|_{\alpha,\beta} \text{d}\gamma.$ One approach may be the Liou-Glaister generalization. Here you initially calculate the partial derivatives point-wise with the EOS with the classical Roe averages (denoted with a tilde symbol). They do not fulfill the first relation and result in a wrong $\Delta p$ (origin of instability). $\Delta p \neq \tilde{p}_{\alpha}\Delta\alpha + \tilde{p}_{\beta}\Delta\beta + \tilde{p}_{\gamma}\Delta\gamma,$ After that you correct them by a projection method (denoted with a hat symbol) $\partial p=- \Delta p + \tilde{p}_{\alpha}\Delta\alpha + \tilde{p}_{\beta}\Delta\beta + \tilde{p}_{\gamma}\Delta\gamma,$ where $\hat{p}_{\alpha} = \tilde{p}_{\alpha} \left(1 + \tilde{p}_{\alpha} \Delta \alpha \partial p / D \right) ,$ $\hat{p}_{\beta} = ...$ $\hat{p}_{\gamma} = ...$ with $D = (\tilde{p}_{\alpha}\Delta\alpha)^2+(\tilde{p}_{\beta}\Delta\beta)^2+(\tilde{p}_{\gamma}\Delta\gamma)^2.$ Now in analogy to that you may approach in a similar way for the preconditioner version by considering the two equations $\text{d}\rho=\left. \frac{\partial{\rho}}{\partial{p}} \right\|_{T} \text{d}p + \left. \frac{\partial{\rho}}{\partial{T}} \right\|_{p} \text{d}T + ...$ and $\text{d}H=\left. \frac{\partial{H}}{\partial{p}} \right\|_{T} \text{d}p + \left. \frac{\partial{H}}{\partial{T}} \right\|_{p} \text{d}T + ...$ to derive consistent primitive partial derivatives. The approach is well described in the paper of Mottura. Regards sbaffini likes this. __________________ Check out my side project: A multiphysics discontinuous Galerkin framework: Youtube, Gitlab. Last edited by Eifoehn4; March 15, 2022 at 14:59.

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