\hypertarget{namespaceplm__functions}{}\section{plm\+\_\+functions Module Reference} \label{namespaceplm__functions}\index{plm\_functions@{plm\_functions}} \subsection{Detailed Description} Piecewise linear reconstruction functions. Date of creation\+: 2008.\+06.\+06 L. White. This module contains routines that handle one-\/dimensionnal finite volume reconstruction using the piecewise linear method (P\+LM). \subsection*{Functions/\+Subroutines} \begin{DoxyCompactItemize} \item real elemental pure function, public \mbox{\hyperlink{namespaceplm__functions_a072affa78922591148b954fa63872246}{plm\+\_\+slope\+\_\+wa}} (h\+\_\+l, h\+\_\+c, h\+\_\+r, h\+\_\+neglect, u\+\_\+l, u\+\_\+c, u\+\_\+r) \begin{DoxyCompactList}\small\item\em Returns a limited P\+LM slope following White and Adcroft, 2008. \mbox{[}units of u\mbox{]} Note that this is not the same as the Colella and Woodward method. \end{DoxyCompactList}\item real elemental pure function, public \mbox{\hyperlink{namespaceplm__functions_a29d2baec39200c5aa448c79181a16f4f}{plm\+\_\+slope\+\_\+cw}} (h\+\_\+l, h\+\_\+c, h\+\_\+r, h\+\_\+neglect, u\+\_\+l, u\+\_\+c, u\+\_\+r) \begin{DoxyCompactList}\small\item\em Returns a limited P\+LM slope following Colella and Woodward 1984. \end{DoxyCompactList}\item real elemental pure function, public \mbox{\hyperlink{namespaceplm__functions_a497e5e73108e08afcb4e2186710ae094}{plm\+\_\+monotonized\+\_\+slope}} (u\+\_\+l, u\+\_\+c, u\+\_\+r, s\+\_\+l, s\+\_\+c, s\+\_\+r) \begin{DoxyCompactList}\small\item\em Returns a limited P\+LM slope following Colella and Woodward 1984. \end{DoxyCompactList}\item real elemental pure function, public \mbox{\hyperlink{namespaceplm__functions_a42fbf62545902eeb6c9e035763496b07}{plm\+\_\+extrapolate\+\_\+slope}} (h\+\_\+l, h\+\_\+c, h\+\_\+neglect, u\+\_\+l, u\+\_\+c) \begin{DoxyCompactList}\small\item\em Returns a P\+LM slope using h2 extrapolation from a cell to the left. Use the negative to extrapolate from the a cell to the right. \end{DoxyCompactList}\item subroutine, public \mbox{\hyperlink{namespaceplm__functions_a0e3a6bedfb3064ed8fae8c8449649054}{plm\+\_\+reconstruction}} (N, h, u, edge\+\_\+values, ppoly\+\_\+coef, h\+\_\+neglect) \begin{DoxyCompactList}\small\item\em Reconstruction by linear polynomials within each cell. \end{DoxyCompactList}\item subroutine, public \mbox{\hyperlink{namespaceplm__functions_a85e83fd8ad314548d861d5048dad8f72}{plm\+\_\+boundary\+\_\+extrapolation}} (N, h, u, edge\+\_\+values, ppoly\+\_\+coef, h\+\_\+neglect) \begin{DoxyCompactList}\small\item\em Reconstruction by linear polynomials within boundary cells. \end{DoxyCompactList}\end{DoxyCompactItemize} \subsection*{Variables} \begin{DoxyCompactItemize} \item \mbox{\Hypertarget{namespaceplm__functions_abc6a0167b8f90d1d1a95c0dec6921053}\label{namespaceplm__functions_abc6a0167b8f90d1d1a95c0dec6921053}} real, parameter \mbox{\hyperlink{namespaceplm__functions_abc6a0167b8f90d1d1a95c0dec6921053}{hneglect\+\_\+dflt}} = 1.E-\/30 \begin{DoxyCompactList}\small\item\em Default negligible cell thickness. \end{DoxyCompactList}\end{DoxyCompactItemize} \subsection{Function/\+Subroutine Documentation} \mbox{\Hypertarget{namespaceplm__functions_a85e83fd8ad314548d861d5048dad8f72}\label{namespaceplm__functions_a85e83fd8ad314548d861d5048dad8f72}} \index{plm\_functions@{plm\_functions}!plm\_boundary\_extrapolation@{plm\_boundary\_extrapolation}} \index{plm\_boundary\_extrapolation@{plm\_boundary\_extrapolation}!plm\_functions@{plm\_functions}} \subsubsection{\texorpdfstring{plm\_boundary\_extrapolation()}{plm\_boundary\_extrapolation()}} {\footnotesize\ttfamily subroutine, public plm\+\_\+functions\+::plm\+\_\+boundary\+\_\+extrapolation (\begin{DoxyParamCaption}\item[{integer, intent(in)}]{N, }\item[{real, dimension(\+:), intent(in)}]{h, }\item[{real, dimension(\+:), intent(in)}]{u, }\item[{real, dimension(\+:,\+:), intent(inout)}]{edge\+\_\+values, }\item[{real, dimension(\+:,\+:), intent(inout)}]{ppoly\+\_\+coef, }\item[{real, intent(in), optional}]{h\+\_\+neglect }\end{DoxyParamCaption})} Reconstruction by linear polynomials within boundary cells. The left and right edge values in the left and right boundary cells, respectively, are estimated using a linear extrapolation within the cells. This extrapolation is E\+X\+A\+CT when the underlying profile is linear. It is assumed that the size of the array \textquotesingle{}u\textquotesingle{} is equal to the number of cells defining \textquotesingle{}grid\textquotesingle{} and \textquotesingle{}ppoly\textquotesingle{}. No consistency check is performed here. \begin{DoxyParams}[1]{Parameters} \mbox{\texttt{ in}} & {\em n} & Number of cells \\ \hline \mbox{\texttt{ in}} & {\em h} & cell widths (size N) \\ \hline \mbox{\texttt{ in}} & {\em u} & cell averages (size N) \\ \hline \mbox{\texttt{ in,out}} & {\em edge\+\_\+values} & edge values of piecewise polynomials, with the same units as u. \\ \hline \mbox{\texttt{ in,out}} & {\em ppoly\+\_\+coef} & coefficients of piecewise polynomials, mainly with the same units as u. \\ \hline \mbox{\texttt{ in}} & {\em h\+\_\+neglect} & A negligibly small width for the purpose of cell reconstructions in the same units as h \\ \hline \end{DoxyParams} Definition at line 269 of file P\+L\+M\+\_\+functions.\+F90. \begin{DoxyCode}{0} \DoxyCodeLine{269 \textcolor{keywordtype}{integer}, \textcolor{keywordtype}{intent(in)} :: N\textcolor{comment}{ !< Number of cells}} \DoxyCodeLine{270 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:)}, \textcolor{keywordtype}{intent(in)} :: h\textcolor{comment}{ !< cell widths (size N)}} \DoxyCodeLine{271 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:)}, \textcolor{keywordtype}{intent(in)} :: u\textcolor{comment}{ !< cell averages (size N)}} \DoxyCodeLine{272 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:,:)}, \textcolor{keywordtype}{intent(inout)} :: edge\_values\textcolor{comment}{ !< edge values of piecewise polynomials,}} \DoxyCodeLine{273 \textcolor{comment}{ !! with the same units as u.}} \DoxyCodeLine{274 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:,:)}, \textcolor{keywordtype}{intent(inout)} :: ppoly\_coef\textcolor{comment}{ !< coefficients of piecewise polynomials, mainly}} \DoxyCodeLine{275 \textcolor{comment}{ !! with the same units as u.}} \DoxyCodeLine{276 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{optional}, \textcolor{keywordtype}{intent(in)} :: h\_neglect\textcolor{comment}{ !< A negligibly small width for}} \DoxyCodeLine{277 \textcolor{comment}{ !! the purpose of cell reconstructions}} \DoxyCodeLine{278 \textcolor{comment}{ !! in the same units as h}} \DoxyCodeLine{279 \textcolor{comment}{! Local variables}} \DoxyCodeLine{280 \textcolor{keywordtype}{ real} :: slope \textcolor{comment}{! retained PLM slope}} \DoxyCodeLine{281 \textcolor{keywordtype}{ real} :: hNeglect} \DoxyCodeLine{282 } \DoxyCodeLine{283 hneglect = hneglect\_dflt ; \textcolor{keywordflow}{if} (\textcolor{keyword}{present}(h\_neglect)) hneglect = h\_neglect} \DoxyCodeLine{284 } \DoxyCodeLine{285 \textcolor{comment}{! Extrapolate from 2 to 1 to estimate slope}} \DoxyCodeLine{286 slope = - plm\_extrapolate\_slope( h(2), h(1), hneglect, u(2), u(1) )} \DoxyCodeLine{287 } \DoxyCodeLine{288 edge\_values(1,1) = u(1) - 0.5 * slope} \DoxyCodeLine{289 edge\_values(1,2) = u(1) + 0.5 * slope} \DoxyCodeLine{290 } \DoxyCodeLine{291 ppoly\_coef(1,1) = edge\_values(1,1)} \DoxyCodeLine{292 ppoly\_coef(1,2) = edge\_values(1,2) - edge\_values(1,1)} \DoxyCodeLine{293 } \DoxyCodeLine{294 \textcolor{comment}{! Extrapolate from N-1 to N to estimate slope}} \DoxyCodeLine{295 slope = plm\_extrapolate\_slope( h(n-1), h(n), hneglect, u(n-1), u(n) )} \DoxyCodeLine{296 } \DoxyCodeLine{297 edge\_values(n,1) = u(n) - 0.5 * slope} \DoxyCodeLine{298 edge\_values(n,2) = u(n) + 0.5 * slope} \DoxyCodeLine{299 } \DoxyCodeLine{300 ppoly\_coef(n,1) = edge\_values(n,1)} \DoxyCodeLine{301 ppoly\_coef(n,2) = edge\_values(n,2) - edge\_values(n,1)} \DoxyCodeLine{302 } \end{DoxyCode} \mbox{\Hypertarget{namespaceplm__functions_a42fbf62545902eeb6c9e035763496b07}\label{namespaceplm__functions_a42fbf62545902eeb6c9e035763496b07}} \index{plm\_functions@{plm\_functions}!plm\_extrapolate\_slope@{plm\_extrapolate\_slope}} \index{plm\_extrapolate\_slope@{plm\_extrapolate\_slope}!plm\_functions@{plm\_functions}} \subsubsection{\texorpdfstring{plm\_extrapolate\_slope()}{plm\_extrapolate\_slope()}} {\footnotesize\ttfamily real elemental pure function, public plm\+\_\+functions\+::plm\+\_\+extrapolate\+\_\+slope (\begin{DoxyParamCaption}\item[{real, intent(in)}]{h\+\_\+l, }\item[{real, intent(in)}]{h\+\_\+c, }\item[{real, intent(in)}]{h\+\_\+neglect, }\item[{real, intent(in)}]{u\+\_\+l, }\item[{real, intent(in)}]{u\+\_\+c }\end{DoxyParamCaption})} Returns a P\+LM slope using h2 extrapolation from a cell to the left. Use the negative to extrapolate from the a cell to the right. \begin{DoxyParams}[1]{Parameters} \mbox{\texttt{ in}} & {\em h\+\_\+l} & Thickness of left cell \mbox{[}units of grid thickness\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em h\+\_\+c} & Thickness of center cell \mbox{[}units of grid thickness\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em h\+\_\+neglect} & A negligible thickness \mbox{[}units of grid thickness\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em u\+\_\+l} & Value of left cell \mbox{[}units of u\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em u\+\_\+c} & Value of center cell \mbox{[}units of u\mbox{]} \\ \hline \end{DoxyParams} Definition at line 161 of file P\+L\+M\+\_\+functions.\+F90. \begin{DoxyCode}{0} \DoxyCodeLine{161 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{intent(in)} :: h\_l\textcolor{comment}{ !< Thickness of left cell [units of grid thickness]}} \DoxyCodeLine{162 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{intent(in)} :: h\_c\textcolor{comment}{ !< Thickness of center cell [units of grid thickness]}} \DoxyCodeLine{163 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{intent(in)} :: h\_neglect\textcolor{comment}{ !< A negligible thickness [units of grid thickness]}} \DoxyCodeLine{164 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{intent(in)} :: u\_l\textcolor{comment}{ !< Value of left cell [units of u]}} \DoxyCodeLine{165 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{intent(in)} :: u\_c\textcolor{comment}{ !< Value of center cell [units of u]}} \DoxyCodeLine{166 \textcolor{comment}{! Local variables}} \DoxyCodeLine{167 \textcolor{keywordtype}{ real} :: left\_edge \textcolor{comment}{! Left edge value [units of u]}} \DoxyCodeLine{168 \textcolor{keywordtype}{ real} :: hl, hc \textcolor{comment}{! Left and central cell thicknesses [units of grid thickness]}} \DoxyCodeLine{169 } \DoxyCodeLine{170 \textcolor{comment}{! Avoid division by zero for vanished cells}} \DoxyCodeLine{171 hl = h\_l + h\_neglect} \DoxyCodeLine{172 hc = h\_c + h\_neglect} \DoxyCodeLine{173 } \DoxyCodeLine{174 \textcolor{comment}{! The h2 scheme is used to compute the left edge value}} \DoxyCodeLine{175 left\_edge = (u\_l*hc + u\_c*hl) / (hl + hc)} \DoxyCodeLine{176 } \DoxyCodeLine{177 plm\_extrapolate\_slope = 2.0 * ( u\_c - left\_edge )} \DoxyCodeLine{178 } \end{DoxyCode} \mbox{\Hypertarget{namespaceplm__functions_a497e5e73108e08afcb4e2186710ae094}\label{namespaceplm__functions_a497e5e73108e08afcb4e2186710ae094}} \index{plm\_functions@{plm\_functions}!plm\_monotonized\_slope@{plm\_monotonized\_slope}} \index{plm\_monotonized\_slope@{plm\_monotonized\_slope}!plm\_functions@{plm\_functions}} \subsubsection{\texorpdfstring{plm\_monotonized\_slope()}{plm\_monotonized\_slope()}} {\footnotesize\ttfamily real elemental pure function, public plm\+\_\+functions\+::plm\+\_\+monotonized\+\_\+slope (\begin{DoxyParamCaption}\item[{real, intent(in)}]{u\+\_\+l, }\item[{real, intent(in)}]{u\+\_\+c, }\item[{real, intent(in)}]{u\+\_\+r, }\item[{real, intent(in)}]{s\+\_\+l, }\item[{real, intent(in)}]{s\+\_\+c, }\item[{real, intent(in)}]{s\+\_\+r }\end{DoxyParamCaption})} Returns a limited P\+LM slope following Colella and Woodward 1984. \begin{DoxyParams}[1]{Parameters} \mbox{\texttt{ in}} & {\em u\+\_\+l} & Value of left cell \mbox{[}units of u\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em u\+\_\+c} & Value of center cell \mbox{[}units of u\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em u\+\_\+r} & Value of right cell \mbox{[}units of u\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em s\+\_\+l} & P\+LM slope of left cell \mbox{[}units of u\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em s\+\_\+c} & P\+LM slope of center cell \mbox{[}units of u\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em s\+\_\+r} & P\+LM slope of right cell \mbox{[}units of u\mbox{]} \\ \hline \end{DoxyParams} Definition at line 122 of file P\+L\+M\+\_\+functions.\+F90. \begin{DoxyCode}{0} \DoxyCodeLine{122 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{intent(in)} :: u\_l\textcolor{comment}{ !< Value of left cell [units of u]}} \DoxyCodeLine{123 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{intent(in)} :: u\_c\textcolor{comment}{ !< Value of center cell [units of u]}} \DoxyCodeLine{124 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{intent(in)} :: u\_r\textcolor{comment}{ !< Value of right cell [units of u]}} \DoxyCodeLine{125 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{intent(in)} :: s\_l\textcolor{comment}{ !< PLM slope of left cell [units of u]}} \DoxyCodeLine{126 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{intent(in)} :: s\_c\textcolor{comment}{ !< PLM slope of center cell [units of u]}} \DoxyCodeLine{127 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{intent(in)} :: s\_r\textcolor{comment}{ !< PLM slope of right cell [units of u]}} \DoxyCodeLine{128 \textcolor{comment}{! Local variables}} \DoxyCodeLine{129 \textcolor{keywordtype}{ real} :: e\_r, e\_l, edge \textcolor{comment}{! Right, left and temporary edge values [units of u]}} \DoxyCodeLine{130 \textcolor{keywordtype}{ real} :: almost\_two \textcolor{comment}{! The number 2, almost.}} \DoxyCodeLine{131 \textcolor{keywordtype}{ real} :: slp \textcolor{comment}{! Magnitude of PLM central slope [units of u]}} \DoxyCodeLine{132 } \DoxyCodeLine{133 almost\_two = 2. * ( 1. - epsilon(s\_c) )} \DoxyCodeLine{134 } \DoxyCodeLine{135 \textcolor{comment}{! Edge values of neighbors abutting this cell}} \DoxyCodeLine{136 e\_r = u\_l + 0.5*s\_l} \DoxyCodeLine{137 e\_l = u\_r - 0.5*s\_r} \DoxyCodeLine{138 slp = abs(s\_c)} \DoxyCodeLine{139 } \DoxyCodeLine{140 \textcolor{comment}{! Check that left edge is between right edge of cell to the left and this cell mean}} \DoxyCodeLine{141 edge = u\_c - 0.5 * s\_c} \DoxyCodeLine{142 \textcolor{keywordflow}{if} ( ( edge - e\_r ) * ( u\_c - edge ) < 0. ) \textcolor{keywordflow}{then}} \DoxyCodeLine{143 edge = 0.5 * ( edge + e\_r )} \DoxyCodeLine{144 slp = min( slp, abs( edge - u\_c ) * almost\_two )} \DoxyCodeLine{145 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{146 } \DoxyCodeLine{147 \textcolor{comment}{! Check that right edge is between left edge of cell to the right and this cell mean}} \DoxyCodeLine{148 edge = u\_c + 0.5 * s\_c} \DoxyCodeLine{149 \textcolor{keywordflow}{if} ( ( edge - u\_c ) * ( e\_l - edge ) < 0. ) \textcolor{keywordflow}{then}} \DoxyCodeLine{150 edge = 0.5 * ( edge + e\_l )} \DoxyCodeLine{151 slp = min( slp, abs( edge - u\_c ) * almost\_two )} \DoxyCodeLine{152 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{153 } \DoxyCodeLine{154 plm\_monotonized\_slope = sign( slp, s\_c )} \DoxyCodeLine{155 } \end{DoxyCode} \mbox{\Hypertarget{namespaceplm__functions_a0e3a6bedfb3064ed8fae8c8449649054}\label{namespaceplm__functions_a0e3a6bedfb3064ed8fae8c8449649054}} \index{plm\_functions@{plm\_functions}!plm\_reconstruction@{plm\_reconstruction}} \index{plm\_reconstruction@{plm\_reconstruction}!plm\_functions@{plm\_functions}} \subsubsection{\texorpdfstring{plm\_reconstruction()}{plm\_reconstruction()}} {\footnotesize\ttfamily subroutine, public plm\+\_\+functions\+::plm\+\_\+reconstruction (\begin{DoxyParamCaption}\item[{integer, intent(in)}]{N, }\item[{real, dimension(\+:), intent(in)}]{h, }\item[{real, dimension(\+:), intent(in)}]{u, }\item[{real, dimension(\+:,\+:), intent(inout)}]{edge\+\_\+values, }\item[{real, dimension(\+:,\+:), intent(inout)}]{ppoly\+\_\+coef, }\item[{real, intent(in), optional}]{h\+\_\+neglect }\end{DoxyParamCaption})} Reconstruction by linear polynomials within each cell. It is assumed that the size of the array \textquotesingle{}u\textquotesingle{} is equal to the number of cells defining \textquotesingle{}grid\textquotesingle{} and \textquotesingle{}ppoly\textquotesingle{}. No consistency check is performed here. \begin{DoxyParams}[1]{Parameters} \mbox{\texttt{ in}} & {\em n} & Number of cells \\ \hline \mbox{\texttt{ in}} & {\em h} & cell widths (size N) \\ \hline \mbox{\texttt{ in}} & {\em u} & cell averages (size N) \\ \hline \mbox{\texttt{ in,out}} & {\em edge\+\_\+values} & edge values of piecewise polynomials, with the same units as u. \\ \hline \mbox{\texttt{ in,out}} & {\em ppoly\+\_\+coef} & coefficients of piecewise polynomials, mainly with the same units as u. \\ \hline \mbox{\texttt{ in}} & {\em h\+\_\+neglect} & A negligibly small width for the purpose of cell reconstructions in the same units as h \\ \hline \end{DoxyParams} Definition at line 187 of file P\+L\+M\+\_\+functions.\+F90. \begin{DoxyCode}{0} \DoxyCodeLine{187 \textcolor{keywordtype}{integer}, \textcolor{keywordtype}{intent(in)} :: N\textcolor{comment}{ !< Number of cells}} \DoxyCodeLine{188 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:)}, \textcolor{keywordtype}{intent(in)} :: h\textcolor{comment}{ !< cell widths (size N)}} \DoxyCodeLine{189 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:)}, \textcolor{keywordtype}{intent(in)} :: u\textcolor{comment}{ !< cell averages (size N)}} \DoxyCodeLine{190 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:,:)}, \textcolor{keywordtype}{intent(inout)} :: edge\_values\textcolor{comment}{ !< edge values of piecewise polynomials,}} \DoxyCodeLine{191 \textcolor{comment}{ !! with the same units as u.}} \DoxyCodeLine{192 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:,:)}, \textcolor{keywordtype}{intent(inout)} :: ppoly\_coef\textcolor{comment}{ !< coefficients of piecewise polynomials, mainly}} \DoxyCodeLine{193 \textcolor{comment}{ !! with the same units as u.}} \DoxyCodeLine{194 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{optional}, \textcolor{keywordtype}{intent(in)} :: h\_neglect\textcolor{comment}{ !< A negligibly small width for}} \DoxyCodeLine{195 \textcolor{comment}{ !! the purpose of cell reconstructions}} \DoxyCodeLine{196 \textcolor{comment}{ !! in the same units as h}} \DoxyCodeLine{197 } \DoxyCodeLine{198 \textcolor{comment}{! Local variables}} \DoxyCodeLine{199 \textcolor{keywordtype}{integer} :: k \textcolor{comment}{! loop index}} \DoxyCodeLine{200 \textcolor{keywordtype}{ real} :: u\_l, u\_c, u\_r \textcolor{comment}{! left, center and right cell averages}} \DoxyCodeLine{201 \textcolor{keywordtype}{ real} :: h\_l, h\_c, h\_r, h\_cn \textcolor{comment}{! left, center and right cell widths}} \DoxyCodeLine{202 \textcolor{keywordtype}{ real} :: slope \textcolor{comment}{! retained PLM slope}} \DoxyCodeLine{203 \textcolor{keywordtype}{ real} :: a, b \textcolor{comment}{! auxiliary variables}} \DoxyCodeLine{204 \textcolor{keywordtype}{ real} :: u\_min, u\_max, e\_l, e\_r, edge} \DoxyCodeLine{205 \textcolor{keywordtype}{ real} :: almost\_one} \DoxyCodeLine{206 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(N)} :: slp, mslp} \DoxyCodeLine{207 \textcolor{keywordtype}{ real} :: hNeglect} \DoxyCodeLine{208 } \DoxyCodeLine{209 hneglect = hneglect\_dflt ; \textcolor{keywordflow}{if} (\textcolor{keyword}{present}(h\_neglect)) hneglect = h\_neglect} \DoxyCodeLine{210 } \DoxyCodeLine{211 almost\_one = 1. - epsilon(slope)} \DoxyCodeLine{212 } \DoxyCodeLine{213 \textcolor{comment}{! Loop on interior cells}} \DoxyCodeLine{214 \textcolor{keywordflow}{do} k = 2,n-1} \DoxyCodeLine{215 slp(k) = plm\_slope\_wa(h(k-1), h(k), h(k+1), hneglect, u(k-1), u(k), u(k+1))} \DoxyCodeLine{216 \textcolor{keywordflow}{ enddo} \textcolor{comment}{! end loop on interior cells}} \DoxyCodeLine{217 } \DoxyCodeLine{218 \textcolor{comment}{! Boundary cells use PCM. Extrapolation is handled after monotonization.}} \DoxyCodeLine{219 slp(1) = 0.} \DoxyCodeLine{220 slp(n) = 0.} \DoxyCodeLine{221 } \DoxyCodeLine{222 \textcolor{comment}{! This loop adjusts the slope so that edge values are monotonic.}} \DoxyCodeLine{223 \textcolor{keywordflow}{do} k = 2, n-1} \DoxyCodeLine{224 mslp(k) = plm\_monotonized\_slope( u(k-1), u(k), u(k+1), slp(k-1), slp(k), slp(k+1) )} \DoxyCodeLine{225 \textcolor{keywordflow}{ enddo} \textcolor{comment}{! end loop on interior cells}} \DoxyCodeLine{226 mslp(1) = 0.} \DoxyCodeLine{227 mslp(n) = 0.} \DoxyCodeLine{228 } \DoxyCodeLine{229 \textcolor{comment}{! Store and return edge values and polynomial coefficients.}} \DoxyCodeLine{230 edge\_values(1,1) = u(1)} \DoxyCodeLine{231 edge\_values(1,2) = u(1)} \DoxyCodeLine{232 ppoly\_coef(1,1) = u(1)} \DoxyCodeLine{233 ppoly\_coef(1,2) = 0.} \DoxyCodeLine{234 \textcolor{keywordflow}{do} k = 2, n-1} \DoxyCodeLine{235 slope = mslp(k)} \DoxyCodeLine{236 u\_l = u(k) - 0.5 * slope \textcolor{comment}{! Left edge value of cell k}} \DoxyCodeLine{237 u\_r = u(k) + 0.5 * slope \textcolor{comment}{! Right edge value of cell k}} \DoxyCodeLine{238 } \DoxyCodeLine{239 edge\_values(k,1) = u\_l} \DoxyCodeLine{240 edge\_values(k,2) = u\_r} \DoxyCodeLine{241 ppoly\_coef(k,1) = u\_l} \DoxyCodeLine{242 ppoly\_coef(k,2) = ( u\_r - u\_l )} \DoxyCodeLine{243 \textcolor{comment}{! Check to see if this evaluation of the polynomial at x=1 would be}} \DoxyCodeLine{244 \textcolor{comment}{! monotonic w.r.t. the next cell's edge value. If not, scale back!}} \DoxyCodeLine{245 edge = ppoly\_coef(k,2) + ppoly\_coef(k,1)} \DoxyCodeLine{246 e\_r = u(k+1) - 0.5 * sign( mslp(k+1), slp(k+1) )} \DoxyCodeLine{247 \textcolor{keywordflow}{if} ( (edge-u(k))*(e\_r-edge)<0.) \textcolor{keywordflow}{then}} \DoxyCodeLine{248 ppoly\_coef(k,2) = ppoly\_coef(k,2) * almost\_one} \DoxyCodeLine{249 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{250 \textcolor{keywordflow}{ enddo}} \DoxyCodeLine{251 edge\_values(n,1) = u(n)} \DoxyCodeLine{252 edge\_values(n,2) = u(n)} \DoxyCodeLine{253 ppoly\_coef(n,1) = u(n)} \DoxyCodeLine{254 ppoly\_coef(n,2) = 0.} \DoxyCodeLine{255 } \end{DoxyCode} \mbox{\Hypertarget{namespaceplm__functions_a29d2baec39200c5aa448c79181a16f4f}\label{namespaceplm__functions_a29d2baec39200c5aa448c79181a16f4f}} \index{plm\_functions@{plm\_functions}!plm\_slope\_cw@{plm\_slope\_cw}} \index{plm\_slope\_cw@{plm\_slope\_cw}!plm\_functions@{plm\_functions}} \subsubsection{\texorpdfstring{plm\_slope\_cw()}{plm\_slope\_cw()}} {\footnotesize\ttfamily real elemental pure function, public plm\+\_\+functions\+::plm\+\_\+slope\+\_\+cw (\begin{DoxyParamCaption}\item[{real, intent(in)}]{h\+\_\+l, }\item[{real, intent(in)}]{h\+\_\+c, }\item[{real, intent(in)}]{h\+\_\+r, }\item[{real, intent(in)}]{h\+\_\+neglect, }\item[{real, intent(in)}]{u\+\_\+l, }\item[{real, intent(in)}]{u\+\_\+c, }\item[{real, intent(in)}]{u\+\_\+r }\end{DoxyParamCaption})} Returns a limited P\+LM slope following Colella and Woodward 1984. \begin{DoxyParams}[1]{Parameters} \mbox{\texttt{ in}} & {\em h\+\_\+l} & Thickness of left cell \mbox{[}units of grid thickness\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em h\+\_\+c} & Thickness of center cell \mbox{[}units of grid thickness\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em h\+\_\+r} & Thickness of right cell \mbox{[}units of grid thickness\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em h\+\_\+neglect} & A negligible thickness \mbox{[}units of grid thickness\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em u\+\_\+l} & Value of left cell \mbox{[}units of u\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em u\+\_\+c} & Value of center cell \mbox{[}units of u\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em u\+\_\+r} & Value of right cell \mbox{[}units of u\mbox{]} \\ \hline \end{DoxyParams} Definition at line 68 of file P\+L\+M\+\_\+functions.\+F90. \begin{DoxyCode}{0} \DoxyCodeLine{68 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{intent(in)} :: h\_l\textcolor{comment}{ !< Thickness of left cell [units of grid thickness]}} \DoxyCodeLine{69 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{intent(in)} :: h\_c\textcolor{comment}{ !< Thickness of center cell [units of grid thickness]}} \DoxyCodeLine{70 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{intent(in)} :: h\_r\textcolor{comment}{ !< Thickness of right cell [units of grid thickness]}} \DoxyCodeLine{71 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{intent(in)} :: h\_neglect\textcolor{comment}{ !< A negligible thickness [units of grid thickness]}} \DoxyCodeLine{72 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{intent(in)} :: u\_l\textcolor{comment}{ !< Value of left cell [units of u]}} \DoxyCodeLine{73 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{intent(in)} :: u\_c\textcolor{comment}{ !< Value of center cell [units of u]}} \DoxyCodeLine{74 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{intent(in)} :: u\_r\textcolor{comment}{ !< Value of right cell [units of u]}} \DoxyCodeLine{75 \textcolor{comment}{! Local variables}} \DoxyCodeLine{76 \textcolor{keywordtype}{ real} :: sigma\_l, sigma\_c, sigma\_r \textcolor{comment}{! Left, central and right slope estimates as}} \DoxyCodeLine{77 \textcolor{comment}{! differences across the cell [units of u]}} \DoxyCodeLine{78 \textcolor{keywordtype}{ real} :: u\_min, u\_max \textcolor{comment}{! Minimum and maximum value across cell [units of u]}} \DoxyCodeLine{79 \textcolor{keywordtype}{ real} :: h\_cn \textcolor{comment}{! Thickness of center cell [units of grid thickness]}} \DoxyCodeLine{80 } \DoxyCodeLine{81 h\_cn = h\_c + h\_neglect} \DoxyCodeLine{82 } \DoxyCodeLine{83 \textcolor{comment}{! Side differences}} \DoxyCodeLine{84 sigma\_r = u\_r - u\_c} \DoxyCodeLine{85 sigma\_l = u\_c - u\_l} \DoxyCodeLine{86 } \DoxyCodeLine{87 \textcolor{comment}{! This is the second order slope given by equation 1.7 of}} \DoxyCodeLine{88 \textcolor{comment}{! Piecewise Parabolic Method, Colella and Woodward (1984),}} \DoxyCodeLine{89 \textcolor{comment}{! http://dx.doi.org/10.1016/0021-991(84)90143-8.}} \DoxyCodeLine{90 \textcolor{comment}{! For uniform resolution it simplifies to ( u\_r - u\_l )/2 .}} \DoxyCodeLine{91 sigma\_c = ( h\_c / ( h\_cn + ( h\_l + h\_r ) ) ) * ( \&} \DoxyCodeLine{92 ( 2.*h\_l + h\_c ) / ( h\_r + h\_cn ) * sigma\_r \&} \DoxyCodeLine{93 + ( 2.*h\_r + h\_c ) / ( h\_l + h\_cn ) * sigma\_l )} \DoxyCodeLine{94 } \DoxyCodeLine{95 \textcolor{comment}{! Limit slope so that reconstructions are bounded by neighbors}} \DoxyCodeLine{96 u\_min = min( u\_l, u\_c, u\_r )} \DoxyCodeLine{97 u\_max = max( u\_l, u\_c, u\_r )} \DoxyCodeLine{98 \textcolor{keywordflow}{if} ( (sigma\_l * sigma\_r) > 0.0 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{99 \textcolor{comment}{! This limits the slope so that the edge values are bounded by the}} \DoxyCodeLine{100 \textcolor{comment}{! two cell averages spanning the edge.}} \DoxyCodeLine{101 plm\_slope\_cw = sign( min( abs(sigma\_c), 2.*min( u\_c - u\_min, u\_max - u\_c ) ), sigma\_c )} \DoxyCodeLine{102 \textcolor{keywordflow}{else}} \DoxyCodeLine{103 \textcolor{comment}{! Extrema in the mean values require a PCM reconstruction avoid generating}} \DoxyCodeLine{104 \textcolor{comment}{! larger extreme values.}} \DoxyCodeLine{105 plm\_slope\_cw = 0.0} \DoxyCodeLine{106 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{107 } \DoxyCodeLine{108 \textcolor{comment}{! This block tests to see if roundoff causes edge values to be out of bounds}} \DoxyCodeLine{109 \textcolor{keywordflow}{if} (u\_c - 0.5*abs(plm\_slope\_cw) < u\_min .or. u\_c + 0.5*abs(plm\_slope\_cw) > u\_max) \textcolor{keywordflow}{then}} \DoxyCodeLine{110 plm\_slope\_cw = plm\_slope\_cw * ( 1. - epsilon(plm\_slope\_cw) )} \DoxyCodeLine{111 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{112 } \DoxyCodeLine{113 \textcolor{comment}{! An attempt to avoid inconsistency when the values become unrepresentable.}} \DoxyCodeLine{114 \textcolor{comment}{! \#\#\# The following 1.E-140 is dimensionally inconsistent. A newer version of}} \DoxyCodeLine{115 \textcolor{comment}{! PLM is progress that will avoid the need for such rounding.}} \DoxyCodeLine{116 \textcolor{keywordflow}{if} (abs(plm\_slope\_cw) < 1.e-140) plm\_slope\_cw = 0.} \DoxyCodeLine{117 } \end{DoxyCode} \mbox{\Hypertarget{namespaceplm__functions_a072affa78922591148b954fa63872246}\label{namespaceplm__functions_a072affa78922591148b954fa63872246}} \index{plm\_functions@{plm\_functions}!plm\_slope\_wa@{plm\_slope\_wa}} \index{plm\_slope\_wa@{plm\_slope\_wa}!plm\_functions@{plm\_functions}} \subsubsection{\texorpdfstring{plm\_slope\_wa()}{plm\_slope\_wa()}} {\footnotesize\ttfamily real elemental pure function, public plm\+\_\+functions\+::plm\+\_\+slope\+\_\+wa (\begin{DoxyParamCaption}\item[{real, intent(in)}]{h\+\_\+l, }\item[{real, intent(in)}]{h\+\_\+c, }\item[{real, intent(in)}]{h\+\_\+r, }\item[{real, intent(in)}]{h\+\_\+neglect, }\item[{real, intent(in)}]{u\+\_\+l, }\item[{real, intent(in)}]{u\+\_\+c, }\item[{real, intent(in)}]{u\+\_\+r }\end{DoxyParamCaption})} Returns a limited P\+LM slope following White and Adcroft, 2008. \mbox{[}units of u\mbox{]} Note that this is not the same as the Colella and Woodward method. \begin{DoxyParams}[1]{Parameters} \mbox{\texttt{ in}} & {\em h\+\_\+l} & Thickness of left cell \mbox{[}units of grid thickness\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em h\+\_\+c} & Thickness of center cell \mbox{[}units of grid thickness\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em h\+\_\+r} & Thickness of right cell \mbox{[}units of grid thickness\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em h\+\_\+neglect} & A negligible thickness \mbox{[}units of grid thickness\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em u\+\_\+l} & Value of left cell \mbox{[}units of u\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em u\+\_\+c} & Value of center cell \mbox{[}units of u\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em u\+\_\+r} & Value of right cell \mbox{[}units of u\mbox{]} \\ \hline \end{DoxyParams} Definition at line 22 of file P\+L\+M\+\_\+functions.\+F90. \begin{DoxyCode}{0} \DoxyCodeLine{22 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{intent(in)} :: h\_l\textcolor{comment}{ !< Thickness of left cell [units of grid thickness]}} \DoxyCodeLine{23 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{intent(in)} :: h\_c\textcolor{comment}{ !< Thickness of center cell [units of grid thickness]}} \DoxyCodeLine{24 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{intent(in)} :: h\_r\textcolor{comment}{ !< Thickness of right cell [units of grid thickness]}} \DoxyCodeLine{25 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{intent(in)} :: h\_neglect\textcolor{comment}{ !< A negligible thickness [units of grid thickness]}} \DoxyCodeLine{26 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{intent(in)} :: u\_l\textcolor{comment}{ !< Value of left cell [units of u]}} \DoxyCodeLine{27 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{intent(in)} :: u\_c\textcolor{comment}{ !< Value of center cell [units of u]}} \DoxyCodeLine{28 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{intent(in)} :: u\_r\textcolor{comment}{ !< Value of right cell [units of u]}} \DoxyCodeLine{29 \textcolor{comment}{! Local variables}} \DoxyCodeLine{30 \textcolor{keywordtype}{ real} :: sigma\_l, sigma\_c, sigma\_r \textcolor{comment}{! Left, central and right slope estimates as}} \DoxyCodeLine{31 \textcolor{comment}{! differences across the cell [units of u]}} \DoxyCodeLine{32 \textcolor{keywordtype}{ real} :: u\_min, u\_max \textcolor{comment}{! Minimum and maximum value across cell [units of u]}} \DoxyCodeLine{33 } \DoxyCodeLine{34 \textcolor{comment}{! Side differences}} \DoxyCodeLine{35 sigma\_r = u\_r - u\_c} \DoxyCodeLine{36 sigma\_l = u\_c - u\_l} \DoxyCodeLine{37 } \DoxyCodeLine{38 \textcolor{comment}{! Quasi-second order difference}} \DoxyCodeLine{39 sigma\_c = 2.0 * ( u\_r - u\_l ) * ( h\_c / ( h\_l + 2.0*h\_c + h\_r + h\_neglect) )} \DoxyCodeLine{40 } \DoxyCodeLine{41 \textcolor{comment}{! Limit slope so that reconstructions are bounded by neighbors}} \DoxyCodeLine{42 u\_min = min( u\_l, u\_c, u\_r )} \DoxyCodeLine{43 u\_max = max( u\_l, u\_c, u\_r )} \DoxyCodeLine{44 \textcolor{keywordflow}{if} ( (sigma\_l * sigma\_r) > 0.0 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{45 \textcolor{comment}{! This limits the slope so that the edge values are bounded by the}} \DoxyCodeLine{46 \textcolor{comment}{! two cell averages spanning the edge.}} \DoxyCodeLine{47 plm\_slope\_wa = sign( min( abs(sigma\_c), 2.*min( u\_c - u\_min, u\_max - u\_c ) ), sigma\_c )} \DoxyCodeLine{48 \textcolor{keywordflow}{else}} \DoxyCodeLine{49 \textcolor{comment}{! Extrema in the mean values require a PCM reconstruction avoid generating}} \DoxyCodeLine{50 \textcolor{comment}{! larger extreme values.}} \DoxyCodeLine{51 plm\_slope\_wa = 0.0} \DoxyCodeLine{52 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{53 } \DoxyCodeLine{54 \textcolor{comment}{! This block tests to see if roundoff causes edge values to be out of bounds}} \DoxyCodeLine{55 \textcolor{keywordflow}{if} (u\_c - 0.5*abs(plm\_slope\_wa) < u\_min .or. u\_c + 0.5*abs(plm\_slope\_wa) > u\_max) \textcolor{keywordflow}{then}} \DoxyCodeLine{56 plm\_slope\_wa = plm\_slope\_wa * ( 1. - epsilon(plm\_slope\_wa) )} \DoxyCodeLine{57 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{58 } \DoxyCodeLine{59 \textcolor{comment}{! An attempt to avoid inconsistency when the values become unrepresentable.}} \DoxyCodeLine{60 \textcolor{comment}{! \#\#\# The following 1.E-140 is dimensionally inconsistent. A newer version of}} \DoxyCodeLine{61 \textcolor{comment}{! PLM is progress that will avoid the need for such rounding.}} \DoxyCodeLine{62 \textcolor{keywordflow}{if} (abs(plm\_slope\_wa) < 1.e-140) plm\_slope\_wa = 0.} \DoxyCodeLine{63 } \end{DoxyCode}