\begin{algorithmic}
\Function{Transport}{$\bold{M}, \boldsymbol{\beta},\bold{g}, \Delta t$} \Comment{Input moments, natural parameters, and gauge parameters for all cells}
\State $\boldsymbol{\beta}\gets \Call{NewtonOptimization}{\bold{M},\boldsymbol{\beta}, \boldsymbol{\phi}(\bold{u} ;\mathbf{g})}$ \Comment{Solve the natural parameters by Alg. \ref{Newton method} in the gauge $\mathbf{g}$}
\State $\bold{F} \gets \Call{ComputeFluxes}{\boldsymbol{\beta}, \boldsymbol{\phi}(\bold{u} ;\mathbf{g})}$\Comment{Compute fluxes of the statistics $\boldsymbol{\phi}(\bold{u} ;\mathbf{g})$ by \eqref{The ME equation:subeq3} }
\State $\bold{M}_{\pm1}, \bold{F}_{\pm1} \gets \Call{SpatialGaugeTransformation}{\bold{M}, \bold{F}, \bold{g}}$\Comment{Perform gauge transformation as in \eqref{The lax F convert} }
\State $\bold{M} \gets \Call{FiniteVolumeStep}{\bold{M}, \bold{F},\bold{M}_{\pm1}, \bold{F}_{\pm1}, \bold{g}, \Delta t}$\Comment{One step forward in time by \eqref{The ME equation fvm 2} }
\State \textbf{return} $\bold{M}, \boldsymbol{\beta}, \bold{g}$
\EndFunction


\Function{Collision}{$\bold{M}, \boldsymbol{\beta},\bold{g}, \Delta t$} \Comment{Input moments, natural parameters, and gauge parameters for all cells}
\State $\bold{M} \gets \Call{SourceTerm}{\bold{M},  \bold{g}, \Delta t}$\Comment{Compute source term by operator splitting \eqref{source update} }
\State \textbf{return} $\bold{M}, \boldsymbol{\beta}, \bold{g}$
\EndFunction


\Function{Step}{$\bold{M}, \boldsymbol{\beta},\bold{g}$} \Comment{Input moments, natural parameters, and gauge parameters for all cells}

\State $\bold{M}, \boldsymbol{\beta}, \bold{g} \gets \Call{Collision}{\bold{M},\boldsymbol{\beta},\bold{g}, \Delta t/2}$\Comment{Compute collision term by operator splitting}
\State $\bold{M}, \boldsymbol{\beta}, \bold{g} \gets \Call{Transport}{\bold{M},\boldsymbol{\beta},\bold{g}, \Delta t}$\Comment{Compute Transport term by operator splitting}
\State $\bold{M}, \boldsymbol{\beta}, \bold{g} \gets \Call{Collision}{\bold{M},\boldsymbol{\beta},\bold{g}, \Delta t/2}$\Comment{Compute collision term by operator splitting}

\State $\bold{g}_H \gets \Call{ComputeGaugeParameters}{\bold{M}, \bold{g}}$\Comment{Compute the Hermite gauge parameters by \eqref{Hermite parameters in moments} }
\State $\bold{M}_H,\boldsymbol{\beta}_H \gets \Call{GaugeTransformation}{\bold{M}, \boldsymbol{\beta}, \bold{g}_H, \bold{g}}$\Comment{Transform into the Hermite gauge using \eqref{Change gauge M F} and \eqref{Maximal Likelihood equation for Exponential Family Distributions: form transformation}}
\State \textbf{return} $\bold{M}_H, \boldsymbol{\beta}_H, \bold{g}_H$
\EndFunction

\end{algorithmic}

\begin{algorithmic}
\Function{Transport}{$\bold{M}, \boldsymbol{\beta},\bold{g}, \Delta t$} \Comment{Input moments, natural parameters, and gauge parameters for all cells}
\State $\boldsymbol{\beta}\gets \Call{NewtonOptimization}{\bold{M},\boldsymbol{\beta}, \boldsymbol{\phi}(\bold{u} ;\mathbf{g})}$ \Comment{Solve the natural parameters by Alg. \ref{Newton method} in the gauge $\mathbf{g}$}
\State $\bold{F} \gets \Call{ComputeFluxes}{\boldsymbol{\beta}, \boldsymbol{\phi}(\bold{u} ;\mathbf{g})}$\Comment{Compute fluxes of the statistics $\boldsymbol{\phi}(\bold{u} ;\mathbf{g})$ by \eqref{The ME equation:subeq3} }
\State $\bold{M}_{\pm1}, \bold{F}_{\pm1} \gets \Call{SpatialGaugeTransformation}{\bold{M}, \bold{F}, \bold{g}}$\Comment{Perform gauge transformation as in \eqref{The lax F convert} }
\State $\bold{M} \gets \Call{FiniteVolumeStep}{\bold{M}, \bold{F},\bold{M}_{\pm1}, \bold{F}_{\pm1}, \bold{g}, \Delta t}$\Comment{One step forward in time by \eqref{The ME equation fvm 2} }
\State \textbf{return} $\bold{M}, \boldsymbol{\beta}, \bold{g}$
\EndFunction
\Function{Collision}{$\bold{M}, \boldsymbol{\beta},\bold{g}, \Delta t$} \Comment{Input moments, natural parameters, and gauge parameters for all cells}
\State $\bold{M} \gets \Call{SourceTerm}{\bold{M}, \bold{g}, \Delta t}$\Comment{Compute source term by operator splitting \eqref{source update} }
\State \textbf{return} $\bold{M}, \boldsymbol{\beta}, \bold{g}$
\EndFunction
\Function{Step}{$\bold{M}, \boldsymbol{\beta},\bold{g}$} \Comment{Input moments, natural parameters, and gauge parameters for all cells}
\State $\bold{M}, \boldsymbol{\beta}, \bold{g} \gets \Call{Collision}{\bold{M},\boldsymbol{\beta},\bold{g}, \Delta t/2}$\Comment{Compute collision term by operator splitting}
\State $\bold{M}, \boldsymbol{\beta}, \bold{g} \gets \Call{Transport}{\bold{M},\boldsymbol{\beta},\bold{g}, \Delta t}$\Comment{Compute Transport term by operator splitting}
\State $\bold{M}, \boldsymbol{\beta}, \bold{g} \gets \Call{Collision}{\bold{M},\boldsymbol{\beta},\bold{g}, \Delta t/2}$\Comment{Compute collision term by operator splitting}
\State $\bold{g}_H \gets \Call{ComputeGaugeParameters}{\bold{M}, \bold{g}}$\Comment{Compute the Hermite gauge parameters by \eqref{Hermite parameters in moments} }
\State $\bold{M}_H,\boldsymbol{\beta}_H \gets \Call{GaugeTransformation}{\bold{M}, \boldsymbol{\beta}, \bold{g}_H, \bold{g}}$\Comment{Transform into the Hermite gauge using \eqref{Change gauge M F} and \eqref{Maximal Likelihood equation for Exponential Family Distributions: form transformation}}
\State \textbf{return} $\bold{M}_H, \boldsymbol{\beta}_H, \bold{g}_H$
\EndFunction
\end{algorithmic}
{
    "title": "stabilizing the maximal entropy moment method for rarefied gas dynamics   at single-precision",
    "id": "2303.02898",
    "abstract": "the maximal entropy moment method (mem) is systematic solution of the challenging problem: generating extended hydrodynamic equations valid for both dense and rarefied gases. however, simulating mem suffers from a computational expensive and ill-conditioned maximal entropy problem. it causes numerical overflow and breakdown when the numerical precision is insufficient, especially for flows like high-speed shock waves. it also prevents modern gpus from accelerating mem with their enormous single floating-point precision computation power. this paper aims to stabilize mem, making it possible to simulating very strong normal shock waves on modern gpus at single precision. we improve the condition number of the maximal entropy problem by proposing gauge transformations, which moves not only flow fields but also hydrodynamic equations into a more optimal coordinate system. we addressed numerical overflow and breakdown in the maximal entropy problem by employing the canonical form of distribution and a modified newton optimization method. moreover, we discovered a counter-intuitive phenomenon that over-refined spatial mesh beyond mean free path degrades the stability of mem. with these techniques, we accomplished single-precision gpu simulations of high speed shock wave up to mach 10 utilizing 35 moments mem, while previous methods only achieved mach 4 on double-precision.",
    "categories": "physics.flu-dyn cs.lg",
    "doi": "",
    "created": "2023-03-06",
    "updated": "2024-02-19",
    "authors": [
        "candi zheng",
        "wang yang",
        "shiyi chen"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2303.02898"
}
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