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