update: auto commit
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@@ -222,6 +222,7 @@
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\newcommand{\form}[1]{\ensuremath{\bm{#1}}}
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\newcommand{\Ric}{\ensuremath{R}}
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\newcommand{\Rscalar}{\ensuremath{R}}
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\newcommand{\curR}{\ensuremath{{R}}}
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\newcommand{\spacecurR}{\ensuremath{\mathcal{R}}}
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\newcommand{\vol}{\ensuremath{\varepsilon}}
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+215
@@ -15,6 +15,182 @@
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\tikzset{zigzag/.style={decorate, decoration=zigzag}}
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\def \L {2.}
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\newcommand{\lapse}{\alpha}
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% \newcommand{\shiftu}[1]{\ensuremath{\tensor{\beta}{^{#1}}}}
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% \newcommand{\shiftd}[1]{\ensuremath{\tensor{\beta}{_{#1}}}}
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\newcommand{\shift}[1]{\tensor{\beta}{#1}}
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\newcommand{\pt}{\Partial{_t}}
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% \newcommand{\csmetric}[1]{\tensor{\tilde{\gamma}}{#1}}
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\NewDocumentCommand{\smetric}{ s m }{% s=可选星号，m=索引们
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\IfBooleanTF{#1}
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{\tensor{\tilde{\gamma}}{#2}}
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{\tensor{\gamma}{#2}}
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}
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\NewDocumentCommand{\Ktf}{ s m }{% s=可选星号，m=索引们
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\IfBooleanTF{#1}
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{\tensor{\tilde{A}}{#2}}
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{\tensor{A}{#2}}
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}
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\NewDocumentCommand{\Gam}{ s m }{% s=可选星号，m=索引们
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\IfBooleanTF{#1}
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{\tensor{\tilde{\Gamma}}{#2}}
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{\tensor{\Gamma}{#2}}
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}
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\NewDocumentCommand{\Gamd}{ s m }{% s=可选星号，m=索引们
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\IfBooleanTF{#1}
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{\tensor{(\tilde{\Gamma}_\text{d})}{#2}}
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{\tensor{(\Gamma_\text{d})}{#2}}
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}
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\NewDocumentCommand{\Rictensor}{ s m }{% s=可选星号，m=索引们
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\IfBooleanTF{#1}
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{\tensor{\tilde{R}}{#2}}
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{\tensor{R}{#2}}
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}
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\RenewDocumentCommand{\spaceD}{ s m }{% s=可选星号，m=索引们
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\IfBooleanTF{#1}
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{\tensor{\tilde{D}}{#2}}
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{\tensor{D}{#2}}
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}
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\NewDocumentCommand{\constraintZ}{ s m }{% s=可选星号，m=索引们
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\IfBooleanTF{#1}
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{\tensor{\tilde{Z}}{#2}}
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{\tensor{Z}{#2}}
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}
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\NewDocumentCommand{\constraintM}{ s m }{% s=可选星号，m=索引们
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\IfBooleanTF{#1}
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{\tensor{\tilde{M}}{#2}}
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{\tensor{M}{#2}}
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}
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\NewDocumentCommand{\dlapse}{ s m }{% s=可选星号，m=索引们
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\IfBooleanTF{#1}
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{\tensor{\tilde{\lapse}}{#2}}
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{\tensor{\lapse}{#2}}
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}
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\NewDocumentCommand{\dshift}{ s m }{% s=可选星号，m=索引们
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\IfBooleanTF{#1}
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{\tensor{\tilde{\beta}}{#2}}
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{\tensor{\beta}{#2}}
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}
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\NewDocumentCommand{\dchi}{ s m }{% s=可选星号，m=索引们
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\IfBooleanTF{#1}
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{\tensor{\tilde{\phi}}{#2}}
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{\tensor{\phi}{#2}}
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}
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\NewDocumentCommand{\dsmetric}{ s m }{% s=可选星号，m=索引们
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\IfBooleanTF{#1}
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{\tensor{\tilde{\psi}}{#2}}
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{\tensor{\psi}{#2}}
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}
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\NewDocumentCommand{\constraintA}{ s m }{% s=可选星号，m=索引们
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\IfBooleanTF{#1}
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{\tensor{\tilde{\mathcal{A}}}{#2}}
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{\tensor{\mathcal{A}}{#2}}
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}
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\NewDocumentCommand{\constraintB}{ s m }{% s=可选星号，m=索引们
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\IfBooleanTF{#1}
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{\tensor{\tilde{\mathcal{B}}}{#2}}
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{\tensor{\mathcal{B}}{#2}}
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}
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\NewDocumentCommand{\constraintC}{ s m }{% s=可选星号，m=索引们
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\IfBooleanTF{#1}
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{\tensor{\tilde{\mathcal{C}}}{#2}}
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{\tensor{\mathcal{C}}{#2}}
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}
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\NewDocumentCommand{\constraintD}{ s m }{% s=可选星号，m=索引们
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\IfBooleanTF{#1}
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{\tensor{\tilde{\mathcal{D}}}{#2}}
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{\tensor{\mathcal{D}}{#2}}
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}
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\newcommand{\oc}[1]{\mathring{#1}}
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\newcommand{\olapse}{\oc{\lapse}}
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% \newcommand{\shiftu}[1]{\ensuremath{\tensor{\beta}{^{#1}}}}
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% \newcommand{\shiftd}[1]{\ensuremath{\tensor{\beta}{_{#1}}}}
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\newcommand{\oshift}[1]{\tensor{{\oc{\beta}}}{#1}}
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% \newcommand{\pt}{\Partial{_t}}
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% % \newcommand{\csmetric}[1]{\tensor{\tilde{\gamma}}{#1}}
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\newcommand{\ochi}{\oc{\chi}}
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\NewDocumentCommand{\osmetric}{ s m }{% s=可选星号，m=索引们
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\IfBooleanTF{#1}
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{\tensor{{\oc{\tilde{\gamma}}}}{#2}}
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{\tensor{{\oc{\gamma}}}{#2}}
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}
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\NewDocumentCommand{\oKtf}{ s m }{% s=可选星号，m=索引们
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\IfBooleanTF{#1}
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{\tensor{{\oc{\tilde{A}}}}{#2}}
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{\tensor{{\oc{A}}}{#2}}
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}
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% \NewDocumentCommand{\Gam}{ s m }{% s=可选星号，m=索引们
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% \IfBooleanTF{#1}
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% {\tensor{\tilde{\Gamma}}{#2}}
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% {\tensor{\Gamma}{#2}}
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% }
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% \NewDocumentCommand{\Gamd}{ s m }{% s=可选星号，m=索引们
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% \IfBooleanTF{#1}
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% {\tensor{(\tilde{\Gamma}_\text{d})}{#2}}
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% {\tensor{(\Gamma_\text{d})}{#2}}
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% }
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% \NewDocumentCommand{\Rictensor}{ s m }{% s=可选星号，m=索引们
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% \IfBooleanTF{#1}
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% {\tensor{\tilde{R}}{#2}}
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% {\tensor{R}{#2}}
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% }
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% \RenewDocumentCommand{\spaceD}{ s m }{% s=可选星号，m=索引们
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% \IfBooleanTF{#1}
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% {\tensor{\tilde{D}}{#2}}
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% {\tensor{D}{#2}}
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% }
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% \NewDocumentCommand{\constraintZ}{ s m }{% s=可选星号，m=索引们
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% \IfBooleanTF{#1}
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% {\tensor{\tilde{Z}}{#2}}
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% {\tensor{Z}{#2}}
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% }
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% \NewDocumentCommand{\constraintM}{ s m }{% s=可选星号，m=索引们
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% \IfBooleanTF{#1}
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% {\tensor{\tilde{M}}{#2}}
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% {\tensor{M}{#2}}
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% }
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% \NewDocumentCommand{\dlapse}{ s m }{% s=可选星号，m=索引们
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% \IfBooleanTF{#1}
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% {\tensor{\tilde{\lapse}}{#2}}
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% {\tensor{\lapse}{#2}}
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% }
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% \NewDocumentCommand{\dshift}{ s m }{% s=可选星号，m=索引们
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% \IfBooleanTF{#1}
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% {\tensor{\tilde{\beta}}{#2}}
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% {\tensor{\beta}{#2}}
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% }
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% \NewDocumentCommand{\dchi}{ s m }{% s=可选星号，m=索引们
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% \IfBooleanTF{#1}
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% {\tensor{\tilde{\phi}}{#2}}
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% {\tensor{\phi}{#2}}
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% }
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\NewDocumentCommand{\odsmetric}{ s m }{% s=可选星号，m=索引们
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\IfBooleanTF{#1}
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{\tensor{\oc{\tilde{\psi}}{}}{#2}}
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{\tensor{\oc{\psi}{}}{#2}}
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}
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% \NewDocumentCommand{\constraintA}{ s m }{% s=可选星号，m=索引们
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% \IfBooleanTF{#1}
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% {\tensor{\tilde{\mathcal{A}}}{#2}}
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% {\tensor{\mathcal{A}}{#2}}
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% }
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% \NewDocumentCommand{\constraintB}{ s m }{% s=可选星号，m=索引们
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% \IfBooleanTF{#1}
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% {\tensor{\tilde{\mathcal{B}}}{#2}}
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% {\tensor{\mathcal{B}}{#2}}
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% }
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% \NewDocumentCommand{\constraintC}{ s m }{% s=可选星号，m=索引们
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% \IfBooleanTF{#1}
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% {\tensor{\tilde{\mathcal{C}}}{#2}}
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% {\tensor{\mathcal{C}}{#2}}
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% }
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% \NewDocumentCommand{\constraintD}{ s m }{% s=可选星号，m=索引们
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% \IfBooleanTF{#1}
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% {\tensor{\tilde{\mathcal{D}}}{#2}}
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% {\tensor{\mathcal{D}}{#2}}
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% }
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\title{Towards Moving Picture Simulations in a Discontinuous Galerkin Framework}
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\subtitle{A Progress Report}
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@@ -479,6 +655,45 @@
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and thus the system is still strongly hyperbolic, and well-posed in the $L^2$ sense.
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\end{frame}
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\begin{frame}{Searching for a first order Z4c}{old workflow: by hand}
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\subsection{old workflow: by hand}
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Here is the Z4c evolution system:
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{\tiny
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\begin{align}
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\pt \chi ={}& \frac{2}{3} \chi \left[ \alpha \left( \hat{K} +
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2\Theta \right) - \spaceD{_i} \shift{^i} \right],\\
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\pt \smetric*{_i_j} ={} & - 2 \lapse \Ktf*{_i_j} + \shift{^k}
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\Partial{_k} \smetric*{_i_j} + 2 \smetric*{_{k(i}} \Partial{_{j)}}
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\shift{^k} - \frac{2}{3} \smetric*{_i_j} \Partial{_k} \shift{^k},\\
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\pt \hat{K} ={}& - \spaceD{^i} \spaceD{_i} \lapse + \lapse \left[
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\Ktf*{_i_j} \Ktf*{^i^j} + \frac{1}{3} \left( \hat{K} + 2 \Theta
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\right)^2 \right] + 4 \pi \lapse \left( S + \rho \right) + \lapse
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\kappa_1 (1-\kappa_2) \Theta + \shift{^i} \Partial{_i} \hat{K},\\
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\pt \Ktf*{_i_j} ={} & \chi \left[ - \spaceD{_i} \spaceD{_j} \lapse
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+ \lapse \left( \Rictensor{_i_j} - 8 \pi \tensor{S}{_i_j} \right)
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\right]^{\text{tf}} + \lapse \left[ \left( \hat{K} + 2 \Theta
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\right) \Ktf*{_i_j} - 2 \Ktf*{^k_i} \Ktf*{_k_j} \right] +
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\shift{^k} \Partial{_k} \Ktf*{_i_j} + 2 \Ktf*{_{k(i}}
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\Partial{_{j)}} \shift{^k} - \frac{2}{3} \Ktf*{_i_j} \Partial{_k}
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\shift{^k},\\
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\pt \Gam*{^i} ={} & - 2 \Ktf*{^i^j} \Partial{_j} \lapse + 2 \lapse
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\left[ \Gam*{^i_j_k} \Ktf*{^j^k} - \frac{3}{2} \Ktf*{^i^j}
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\Partial{_j} \ln(\chi) - \frac{1}{3} \smetric*{^i^j} \Partial{_j}
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\left( 2 \hat{K} + \Theta \right) - 8 \pi \smetric*{^i^j}
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\tensor{S}{_j} \right] + \smetric*{^j^k} \Partial{_j} \Partial{_k}
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\shift{^i} + \frac{1}{3} \smetric*{^i^j} \Partial{_j} \Partial{_k}
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\shift{^k} \notag\\
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& {} + \shift{^j} \Partial{_j} \Gam*{^i} - \Gamd*{^j} \Partial{_j}
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\shift{^i} + \frac{2}{3} \Gamd*{^i} \Partial{_j} \shift{^j} - 2
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\lapse \kappa_1 \left[ \Gam*{^i} - \Gamd*{^i} \right], \\
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\pt \Theta ={} & \frac{1}{2} \lapse \left[ \Rscalar - \Ktf*{_i_j}
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\Ktf*{^i^j} + \frac{2}{3} \left( \hat{K} + 2 \Theta \right)^2
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\right] - \lapse \left[ 8 \pi \rho + \kappa_1 (2+\kappa_2) \Theta
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\right] + \shift{^i} \Partial{_i} \Theta,
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\end{align}
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}
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\end{frame}
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\begin{frame}{References}
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\printbibliography
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\end{frame}
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