Commit 054eb4c4 by Philippe Helluy

### end review 1

parent bb5dac00
 ... ... @@ -1237,7 +1237,10 @@ the transport operator then reduces to a simple shift. Before the collision step, we have thus \begin_inset Formula \v f_{1,i,j}^{n+1,-}=\v f_{1,i+1,j}^{n},\quad\v f_{2,i,j}^{n+1,-}=\v f_{2,i-1,j}^{n},\quad\v f_{3,i,j}^{n+1,-}=\v f_{3,i,j+1}^{n},\quad\v f_{4,i,j}^{n+1,-}=\v f_{4,i,j-1}^{n}.\label{eq:shift-algorithm} \begin{array}{c} \v f_{1,i,j}^{n+1,-}=\v f_{1,i+1,j}^{n},\v f_{2,i,j}^{n+1,-}=\v f_{2,i-1,j}^{n},\\ \v f_{3,i,j}^{n+1,-}=\v f_{3,i,j+1}^{n},\quad\v f_{4,i,j}^{n+1,-}=\v f_{4,i,j-1}^{n}. \end{array}\label{eq:shift-algorithm} \end_inset ... ...
 ... ... @@ -75,8 +75,9 @@ \begin_body \begin_layout Standard Answers to reviewer 1 (most of the changes are highlighted in green) \begin_layout Subsection* Answers to reviewer 1 (most of the corresponding changes are highlighted in green) \end_layout \begin_layout Enumerate ... ... @@ -196,29 +197,36 @@ One of the authors deeply studied Riemann problems for MHD with Godunov \begin_inset Newline newline \end_inset We have written an additional remark at the end of section 3.2 \end_layout \begin_layout Enumerate In the numerical tests you choose epsilon=10^{-3}. What happens for different scales of this value? \begin_inset Newline newline \end_inset We have not tried other values of this parameter. epsilon has been fixed to this value for comparison with physics papers. The objective here is really to catch the linear behavior of the instability. \end_layout \begin_layout Standard Other comments: \end_layout \begin_layout Enumerate \begin_inset Quotes eld \end_inset Please write MagnetoHydroDynamic extensively at the beginning of the paper and then replace it by the acronym. The same applies to GPU (in the abstract is used the acronym, but never introduced). \begin_inset Quotes erd \end_inset \begin_layout Standard We have corrected all the signaled typos. Thanks for pointing then to us. \end_layout \begin_layout Subsection* Answers to reviewer 2 (most of the corresponding changes are highlighted in red) \end_layout \begin_inset Newline newline \end_inset \begin_layout Standard We have corrected the writing at several places. \end_layout \end_body ... ...
 ... ... @@ -10,6 +10,17 @@ year={2020} } @article{bouchut2010multiwave, title={A multiwave approximate Riemann solver for ideal MHD based on relaxation II: numerical implementation with 3 and 5 waves}, author={Bouchut, Fran{\c{c}}ois and Klingenberg, Christian and Waagan, Knut}, journal={Numerische Mathematik}, volume={115}, number={4}, pages={647--679}, year={2010}, publisher={Springer} } @article{klockner2012pycuda, title={PyCUDA and PyOpenCL: A scripting-based approach to GPU run-time code generation}, author={Kl{\"o}ckner, Andreas and Pinto, Nicolas and Lee, Yunsup and Catanzaro, Bryan and Ivanov, Paul and Fasih, Ahmed}, ... ...
 ... ... @@ -106,7 +106,7 @@ \abstract{This paper is devoted to the simulation of \revA{Magnetohydrohynamics} flows with complex structures. This kind flows present instabilities that generate shock waves. We propose a robust and precise numerical method based on the waves. We propose a robust and accurate numerical method based on the Lattice Boltzmann methodology. We explain how to adjust the numerical viscosity in order to obtain stable, \revA{accurate results in smooth or discontinuous parts of the flow} and reduced divergence errors. This method can handle shock wave and is almost second order. ... ... @@ -148,7 +148,7 @@ on a regular grid. An important aspect of the LBM is the choice of the relaxation parameter in the collision step. The choice of the parameter allows adjusting the numerical viscosity of the LBM scheme. We provide a analysis in a simplified one-dimensional framework which We provide an analysis in a simplified one-dimensional framework which shows that it is possible to adjust more precisely the numerical viscosity with a generalized matrix relaxation parameter. We also show that the divergence cleaning effect is improved if the relaxation parameter ... ... @@ -237,9 +237,9 @@ $\v u=(u_{1},u_{2},u_{3})^{T},\quad\v B=(B_{1},B_{2},B_{3})^{T},$ the pressure is defined by the pressure is given \revA{by a perfect-gas law with a constant polytropic exponent $\gamma>1$ } $p=(\gamma-1)(Q-\rho\frac{\v u\cdot\v u}{2}-\frac{\v B\cdot\v B}{2}). p=(\gamma-1)(Q-\rho\frac{\v u\cdot\v u}{2}-\frac{\v B\cdot\v B}{2}),$ The other variables are the density $\rho$, the total energy $Q$, ... ... @@ -424,7 +424,7 @@ of the original equations. See for instance \cite{dellar2013interpretation}). See also below for a mathematical analysis in the one-dimensional case. \section{vide} %\section{vide} \section{Numerical method} ... ... @@ -474,8 +474,11 @@  the transport operator then reduces to a simple shift. Before the collision step, we have thus \v f_{1,i,j}^{n+1,-}=\v f_{1,i+1,j}^{n},\quad\v f_{2,i,j}^{n+1,-}=\v f_{2,i-1,j}^{n},\quad\v f_{3,i,j}^{n+1,-}=\v f_{3,i,j+1}^{n},\quad\v f_{4,i,j}^{n+1,-}=\v f_{4,i,j-1}^{n}.\label{eq:shift-algorithm} \label{eq:shift-algorithm}\left\{ \begin{array}{l} \v f_{1,i,j}^{n+1,-}=\v f_{1,i+1,j}^{n},\v f_{2,i,j}^{n+1,-}=\v f_{2,i-1,j}^{n},\\ \v f_{3,i,j}^{n+1,-}=\v f_{3,i,j+1}^{n},\quad\v f_{4,i,j}^{n+1,-}=\v f_{4,i,j-1}^{n}. \end{array}\right. ... ... @@ -490,7 +493,7 @@ $\v f_{k,i,j}^{n+1}=\v f_{k}^{eq}(\v w_{i,j}^{n+1}).\label{eq:equilibrium} As stated above, it is more precise to consider an over-relaxation As stated above, it is more accurate to consider an over-relaxation $\v f_{k,i,j}^{n+1}=2\v f_{k}^{eq}(\v w_{i,j}^{n+1})-\v f_{k,i,j}^{n+1,-}.$ ... ... @@ -505,6 +508,7 @@$ where $\omega=\frac{2\Delta t}{2\tau+\Delta t}$. Typically, one will choose $\omega=1.9$ in our simulations. \revA{We see that the whole algorithm is extremely simple. It is a succession of "shifts" (\ref{eq:shift-algorithm}) and "collisions" (\ref{eq:collision}. For $\omega=2$ the scheme is second order but unstable in shocks. For $\omega=1$ the scheme is very robust, entropy dissipative, but quite diffusive. It would be interesting to construct a rigorous strategy for choosing locally the optimal value of $\omega$. Let us mention that many finite volume schemes have been designed for solving MHD equations, with specific Riemann solvers. We can mention for instance the solver of \cite{bouchut2010multiwave}, based on a relaxation approach, with proven stability and accuracy features. However, the solver presented in \cite{bouchut2010multiwave} is more complicated to program and less computationally efficient. We can thus compensate the slightly lower accuracy of the kinetic scheme by finer meshes, without increasing too much the computational load.} \subsection{Boundary conditions} ... ... @@ -531,7 +535,7 @@ $For the analysis, it is possible to replace the scalar relaxation parameter$\omega$by a matrix$\vom$, for more generality. In the following, we understand the comparison of matrices in the usual way, following, we \revA{establish the comparison between matrices} in the usual way, by the comparison of the associated quadratic form (the resulting order is thus not total). ... ... @@ -581,7 +585,7 @@$ In order to check practically the accuracy of the approximation, we apply the above analysis for a simplified system of two conservation laws laws. We consider the one-dimensional isothermal Euler equations with a diagonal diffusion of $\epsilon\partial_{xx}\left(\rho,\rho u\right)^{T}$. ... ... @@ -597,7 +601,7 @@ \end{align} This system is a very simplified version of the MHD equation, where the magnetic field is assumed to vanish and the gas is supposed to the magnetic field is assumed to vanish and the gas is supposed to be isothermal. We have chosen to use this specific non-physical diffusion for our first test since it is exactly the type of diffusion that is apparent in a standard finite volume code using a Lax-Friedrichs ... ... @@ -635,15 +639,15 @@ For our first test of the matrix relaxation, we solve (\ref{eq: Isothermal Navier-Stokes}) comparing the Lattice Boltzmann sheme using (\ref{eq:relax-with-diffusion}) as relaxation matrix and a standard explicit centered finite volume scheme for approximating (\eqref{eq: Isothermal Navier-Stokes}). scheme for approximating (\ref{eq: Isothermal Navier-Stokes}). In this centered scheme, the time step is taken very small in such way that the stability condition is satisfied and that the time integration error can be neglected. In other words, the equivalent PDE of both schemes is (\ref{eq:equiv-eq-2order}). Hence, given the parameters in both schemes are set to represent the same diffusion $\epsilon$, one should get the same type of diffusion for both schemes.\\ To test this we take the simple test case of a stationary viscous shock:\\ To test this we take the simple case of a stationary viscous shock.\\ The initial data for this shock tube problem are \begin{align} \wvec_{l}=\left(\rho_{l},\rho_{l}u_{l}\right)^{T}\ \ \ \wvec_{r}=\left(\frac{\rho_{l}u_{l}^{2}}{c},\frac{c}{u_{l}}\right)^{T} ... ... @@ -657,7 +661,7 @@ $scheme we set$\lambda=40$. The results on a$128$cell grid at time$T=0.1$are shown in Figure \ref{Fg: ViscouseShock}. The test shows that the matrix relaxation lattice Boltzmann scheme provides the correct diffusion and therefore results in the right viscose profile. the correct diffusion and therefore results in the right viscous profile. \begin{figure}[h!] \centering{}\includegraphics[width=0.4\textwidth]{png/FVvsMatRel}\includegraphics[width=0.4\textwidth]{png/FVvsMatRelZoom} \caption{{Viscous Shock Test with$\epsilon=0.1$: Comparison of the viscous ... ... @@ -1068,7 +1072,7 @@$ 0 \end{matrix}\right), \] with $\epsilon=1.0\cdot10^{-3}$. with $\epsilon=10^{-3}$. The asymptotic magnetic field strength for large $r$ is unity, and thus defines our normalization. We also point out that we consider ... ... @@ -1212,7 +1216,6 @@ $of the kinetic energy growth rate according to the grid and the order of the numerical method. The converged value for the growth rate is near 1.3 for an initial perturbation$\epsilon=1.0\cdot10^{-3}$. In The simulations of the tilt instability that we perform are post-processed at regular time intervals. In Figures \ref{fig:kinetic-1} and \ref{fig:kinetic-2}, ... ... @@ -1231,9 +1234,9 @@$ The converged growth rate is close to 1.45 for our simulations. It is higher than the results of \cite{richard1990tilt} and \cite{lankalapalli2007adaptive}. In \cite{keppens2014interacting} for a similar configuration the growth rate is evaluated to 1.498. This probably very close to the exact rate because the simulation is conducted with a very precise adaptive scheme. Our results seem thus to be quite precise on the growth rate is evaluated to 1.498. This is probably very close to the exact rate because the simulation is conducted with a very accurate adaptive scheme. Our results seem thus to be quite correct on the finest mesh. %\tikzsetnextfilename{kinetic_512-dirichlet} ... ... @@ -1406,6 +1409,6 @@ \$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bibliographystyle{plain} \bibliography{mhd_lbm.bib} \bibliography{spring_mhd_lbm.bib} \end{document}
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