\hypertarget{namespaceppm__functions}{}\section{ppm\+\_\+functions Module Reference} \label{namespaceppm__functions}\index{ppm\+\_\+functions@{ppm\+\_\+functions}} \subsection{Detailed Description} Provides functions used with the Piecewise-\/\+Parabolic-\/\+Method in the vertical A\+LE algorithm. \subsection*{Functions/\+Subroutines} \begin{DoxyCompactItemize} \item subroutine, public \hyperlink{namespaceppm__functions_a5c4e47be75ad413904902582b3d0d4a4}{ppm\+\_\+reconstruction} (N, h, u, edge\+\_\+values, ppoly\+\_\+coef, h\+\_\+neglect, answers\+\_\+2018) \begin{DoxyCompactList}\small\item\em Builds quadratic polynomials coefficients from cell mean and edge values. \end{DoxyCompactList}\item subroutine \hyperlink{namespaceppm__functions_a3d365f4992a6d3b5cbcaca630c502b95}{ppm\+\_\+limiter\+\_\+standard} (N, h, u, edge\+\_\+values, h\+\_\+neglect, answers\+\_\+2018) \begin{DoxyCompactList}\small\item\em Adjusts edge values using the standard P\+PM limiter (Colella \& Woodward, J\+CP 1984) after first checking that the edge values are bounded by neighbors cell averages and that the edge values are monotonic between cell averages. \end{DoxyCompactList}\item subroutine, public \hyperlink{namespaceppm__functions_ac8f382eb9d1f0df80aab9e07eb2ceebe}{ppm\+\_\+boundary\+\_\+extrapolation} (N, h, u, edge\+\_\+values, ppoly\+\_\+coef, h\+\_\+neglect) \begin{DoxyCompactList}\small\item\em Reconstruction by parabolas within boundary cells. \end{DoxyCompactList}\end{DoxyCompactItemize} \subsection*{Variables} \begin{DoxyCompactItemize} \item real, parameter \hyperlink{namespaceppm__functions_a86e3d1a796b4934167d563eda5310a0e}{hneglect\+\_\+dflt} = 1.E-\/30 \begin{DoxyCompactList}\small\item\em A tiny width that is so small that adding it to cell widths does not change the value due to a computational representation. It is used to avoid division by zero. \end{DoxyCompactList}\end{DoxyCompactItemize} \subsection{Function/\+Subroutine Documentation} \mbox{\Hypertarget{namespaceppm__functions_ac8f382eb9d1f0df80aab9e07eb2ceebe}\label{namespaceppm__functions_ac8f382eb9d1f0df80aab9e07eb2ceebe}} \index{ppm\+\_\+functions@{ppm\+\_\+functions}!ppm\+\_\+boundary\+\_\+extrapolation@{ppm\+\_\+boundary\+\_\+extrapolation}} \index{ppm\+\_\+boundary\+\_\+extrapolation@{ppm\+\_\+boundary\+\_\+extrapolation}!ppm\+\_\+functions@{ppm\+\_\+functions}} \subsubsection{\texorpdfstring{ppm\+\_\+boundary\+\_\+extrapolation()}{ppm\_boundary\_extrapolation()}} {\footnotesize\ttfamily subroutine, public ppm\+\_\+functions\+::ppm\+\_\+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 parabolas within boundary cells. \begin{DoxyParams}[1]{Parameters} \mbox{\tt in} & {\em n} & Number of cells\\ \hline \mbox{\tt in} & {\em h} & cell widths (size N) \mbox{[}H\mbox{]}\\ \hline \mbox{\tt in} & {\em u} & cell averages (size N) \mbox{[}A\mbox{]}\\ \hline \mbox{\tt in,out} & {\em edge\+\_\+values} & edge values of piecewise polynomials \mbox{[}A\mbox{]}\\ \hline \mbox{\tt in,out} & {\em ppoly\+\_\+coef} & coefficients of piecewise polynomials, mainly \mbox{[}A\mbox{]}\\ \hline \mbox{\tt in} & {\em h\+\_\+neglect} & A negligibly small width for the purpose of cell reconstructions \mbox{[}H\mbox{]} \\ \hline \end{DoxyParams} Definition at line 134 of file P\+P\+M\+\_\+functions.\+F90. \begin{DoxyCode} 134 \textcolor{comment}{!------------------------------------------------------------------------------} 135 \textcolor{comment}{! Reconstruction by parabolas within boundary cells.} 136 \textcolor{comment}{!} 137 \textcolor{comment}{! The following explanations apply to the left boundary cell. The same} 138 \textcolor{comment}{! reasoning holds for the right boundary cell.} 139 \textcolor{comment}{!} 140 \textcolor{comment}{! A parabola needs to be built in the cell and requires three degrees of} 141 \textcolor{comment}{! freedom, which are the right edge value and slope and the cell average.} 142 \textcolor{comment}{! The right edge values and slopes are taken to be that of the neighboring} 143 \textcolor{comment}{! cell (i.e., the left edge value and slope of the neighboring cell).} 144 \textcolor{comment}{! The resulting parabola is not necessarily monotonic and the traditional} 145 \textcolor{comment}{! PPM limiter is used to modify one of the edge values in order to yield} 146 \textcolor{comment}{! a monotonic parabola.} 147 \textcolor{comment}{!} 148 \textcolor{comment}{! N: number of cells in grid} 149 \textcolor{comment}{! h: thicknesses of grid cells} 150 \textcolor{comment}{! u: cell averages to use in constructing piecewise polynomials} 151 \textcolor{comment}{! edge\_values : edge values of piecewise polynomials} 152 \textcolor{comment}{! ppoly\_coef : coefficients of piecewise polynomials} 153 \textcolor{comment}{!} 154 \textcolor{comment}{! It is assumed that the size of the array 'u' is equal to the number of cells} 155 \textcolor{comment}{! defining 'grid' and 'ppoly'. No consistency check is performed here.} 156 \textcolor{comment}{!------------------------------------------------------------------------------} 157 158 \textcolor{comment}{! Arguments} 159 \textcolor{keywordtype}{integer}, \textcolor{keywordtype}{intent(in)} :: n\textcolor{comment}{ !< Number of cells} 160 \textcolor{keywordtype}{real}, \textcolor{keywordtype}{dimension(:)}, \textcolor{keywordtype}{intent(in)} :: h\textcolor{comment}{ !< cell widths (size N) [H]} 161 \textcolor{keywordtype}{real}, \textcolor{keywordtype}{dimension(:)}, \textcolor{keywordtype}{intent(in)} :: u\textcolor{comment}{ !< cell averages (size N) [A]} 162 \textcolor{keywordtype}{real}, \textcolor{keywordtype}{dimension(:,:)}, \textcolor{keywordtype}{intent(inout)} :: edge\_values\textcolor{comment}{ !< edge values of piecewise polynomials [A]} 163 \textcolor{keywordtype}{real}, \textcolor{keywordtype}{dimension(:,:)}, \textcolor{keywordtype}{intent(inout)} :: ppoly\_coef\textcolor{comment}{ !< coefficients of piecewise polynomials, mainly [A]} 164 \textcolor{keywordtype}{real}, \textcolor{keywordtype}{optional}, \textcolor{keywordtype}{intent(in)} :: h\_neglect\textcolor{comment}{ !< A negligibly small width for} 165 \textcolor{comment}{ !! the purpose of cell reconstructions [H]} 166 167 \textcolor{comment}{! Local variables} 168 \textcolor{keywordtype}{integer} :: i0, i1 169 \textcolor{keywordtype}{real} :: u0, u1 170 \textcolor{keywordtype}{real} :: h0, h1 171 \textcolor{keywordtype}{real} :: a, b, c 172 \textcolor{keywordtype}{real} :: u0\_l, u0\_r 173 \textcolor{keywordtype}{real} :: u1\_l, u1\_r 174 \textcolor{keywordtype}{real} :: slope 175 \textcolor{keywordtype}{real} :: exp1, exp2 176 \textcolor{keywordtype}{real} :: hneglect 177 178 hneglect = hneglect\_dflt ; \textcolor{keywordflow}{if} (\textcolor{keyword}{present}(h\_neglect)) hneglect = h\_neglect 179 180 \textcolor{comment}{! ----- Left boundary -----} 181 i0 = 1 182 i1 = 2 183 h0 = h(i0) 184 h1 = h(i1) 185 u0 = u(i0) 186 u1 = u(i1) 187 188 \textcolor{comment}{! Compute the left edge slope in neighboring cell and express it in} 189 \textcolor{comment}{! the global coordinate system} 190 b = ppoly\_coef(i1,2) 191 u1\_r = b *((h0+hneglect)/(h1+hneglect)) \textcolor{comment}{! derivative evaluated at xi = 0.0,} 192 \textcolor{comment}{! expressed w.r.t. xi (local coord. system)} 193 194 \textcolor{comment}{! Limit the right slope by the PLM limited slope} 195 slope = 2.0 * ( u1 - u0 ) 196 \textcolor{keywordflow}{if} ( abs(u1\_r) > abs(slope) ) \textcolor{keywordflow}{then} 197 u1\_r = slope 198 \textcolor{keywordflow}{ endif} 199 200 \textcolor{comment}{! The right edge value in the boundary cell is taken to be the left} 201 \textcolor{comment}{! edge value in the neighboring cell} 202 u0\_r = edge\_values(i1,1) 203 204 \textcolor{comment}{! Given the right edge value and slope, we determine the left} 205 \textcolor{comment}{! edge value and slope by computing the parabola as determined by} 206 \textcolor{comment}{! the right edge value and slope and the boundary cell average} 207 u0\_l = 3.0 * u0 + 0.5 * u1\_r - 2.0 * u0\_r 208 209 \textcolor{comment}{! Apply the traditional PPM limiter} 210 exp1 = (u0\_r - u0\_l) * (u0 - 0.5*(u0\_l+u0\_r)) 211 exp2 = (u0\_r - u0\_l) * (u0\_r - u0\_l) / 6.0 212 213 \textcolor{keywordflow}{if} ( exp1 > exp2 ) \textcolor{keywordflow}{then} 214 u0\_l = 3.0 * u0 - 2.0 * u0\_r 215 \textcolor{keywordflow}{ endif} 216 217 \textcolor{keywordflow}{if} ( exp1 < -exp2 ) \textcolor{keywordflow}{then} 218 u0\_r = 3.0 * u0 - 2.0 * u0\_l 219 \textcolor{keywordflow}{ endif} 220 221 edge\_values(i0,1) = u0\_l 222 edge\_values(i0,2) = u0\_r 223 224 a = u0\_l 225 b = 6.0 * u0 - 4.0 * u0\_l - 2.0 * u0\_r 226 c = 3.0 * ( u0\_r + u0\_l - 2.0 * u0 ) 227 228 ppoly\_coef(i0,1) = a 229 ppoly\_coef(i0,2) = b 230 ppoly\_coef(i0,3) = c 231 232 \textcolor{comment}{! ----- Right boundary -----} 233 i0 = n-1 234 i1 = n 235 h0 = h(i0) 236 h1 = h(i1) 237 u0 = u(i0) 238 u1 = u(i1) 239 240 \textcolor{comment}{! Compute the right edge slope in neighboring cell and express it in} 241 \textcolor{comment}{! the global coordinate system} 242 b = ppoly\_coef(i0,2) 243 c = ppoly\_coef(i0,3) 244 u1\_l = (b + 2*c) \textcolor{comment}{! derivative evaluated at xi = 1.0} 245 u1\_l = u1\_l * ((h1+hneglect)/(h0+hneglect)) 246 247 \textcolor{comment}{! Limit the left slope by the PLM limited slope} 248 slope = 2.0 * ( u1 - u0 ) 249 \textcolor{keywordflow}{if} ( abs(u1\_l) > abs(slope) ) \textcolor{keywordflow}{then} 250 u1\_l = slope 251 \textcolor{keywordflow}{ endif} 252 253 \textcolor{comment}{! The left edge value in the boundary cell is taken to be the right} 254 \textcolor{comment}{! edge value in the neighboring cell} 255 u0\_l = edge\_values(i0,2) 256 257 \textcolor{comment}{! Given the left edge value and slope, we determine the right} 258 \textcolor{comment}{! edge value and slope by computing the parabola as determined by} 259 \textcolor{comment}{! the left edge value and slope and the boundary cell average} 260 u0\_r = 3.0 * u1 - 0.5 * u1\_l - 2.0 * u0\_l 261 262 \textcolor{comment}{! Apply the traditional PPM limiter} 263 exp1 = (u0\_r - u0\_l) * (u1 - 0.5*(u0\_l+u0\_r)) 264 exp2 = (u0\_r - u0\_l) * (u0\_r - u0\_l) / 6.0 265 266 \textcolor{keywordflow}{if} ( exp1 > exp2 ) \textcolor{keywordflow}{then} 267 u0\_l = 3.0 * u1 - 2.0 * u0\_r 268 \textcolor{keywordflow}{ endif} 269 270 \textcolor{keywordflow}{if} ( exp1 < -exp2 ) \textcolor{keywordflow}{then} 271 u0\_r = 3.0 * u1 - 2.0 * u0\_l 272 \textcolor{keywordflow}{ endif} 273 274 edge\_values(i1,1) = u0\_l 275 edge\_values(i1,2) = u0\_r 276 277 a = u0\_l 278 b = 6.0 * u1 - 4.0 * u0\_l - 2.0 * u0\_r 279 c = 3.0 * ( u0\_r + u0\_l - 2.0 * u1 ) 280 281 ppoly\_coef(i1,1) = a 282 ppoly\_coef(i1,2) = b 283 ppoly\_coef(i1,3) = c 284 \end{DoxyCode} \mbox{\Hypertarget{namespaceppm__functions_a3d365f4992a6d3b5cbcaca630c502b95}\label{namespaceppm__functions_a3d365f4992a6d3b5cbcaca630c502b95}} \index{ppm\+\_\+functions@{ppm\+\_\+functions}!ppm\+\_\+limiter\+\_\+standard@{ppm\+\_\+limiter\+\_\+standard}} \index{ppm\+\_\+limiter\+\_\+standard@{ppm\+\_\+limiter\+\_\+standard}!ppm\+\_\+functions@{ppm\+\_\+functions}} \subsubsection{\texorpdfstring{ppm\+\_\+limiter\+\_\+standard()}{ppm\_limiter\_standard()}} {\footnotesize\ttfamily subroutine ppm\+\_\+functions\+::ppm\+\_\+limiter\+\_\+standard (\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, intent(in), optional}]{h\+\_\+neglect, }\item[{logical, intent(in), optional}]{answers\+\_\+2018 }\end{DoxyParamCaption})\hspace{0.3cm}{\ttfamily [private]}} Adjusts edge values using the standard P\+PM limiter (Colella \& Woodward, J\+CP 1984) after first checking that the edge values are bounded by neighbors cell averages and that the edge values are monotonic between cell averages. \begin{DoxyParams}[1]{Parameters} \mbox{\tt in} & {\em n} & Number of cells\\ \hline \mbox{\tt in} & {\em h} & cell widths (size N) \mbox{[}H\mbox{]}\\ \hline \mbox{\tt in} & {\em u} & cell average properties (size N) \mbox{[}A\mbox{]}\\ \hline \mbox{\tt in,out} & {\em edge\+\_\+values} & Potentially modified edge values \mbox{[}A\mbox{]}\\ \hline \mbox{\tt in} & {\em h\+\_\+neglect} & A negligibly small width \mbox{[}H\mbox{]}\\ \hline \mbox{\tt in} & {\em answers\+\_\+2018} & If true use older, less acccurate expressions. \\ \hline \end{DoxyParams} Definition at line 63 of file P\+P\+M\+\_\+functions.\+F90. \begin{DoxyCode} 63 \textcolor{keywordtype}{integer}, \textcolor{keywordtype}{intent(in)} :: n\textcolor{comment}{ !< Number of cells} 64 \textcolor{keywordtype}{real}, \textcolor{keywordtype}{dimension(:)}, \textcolor{keywordtype}{intent(in)} :: h\textcolor{comment}{ !< cell widths (size N) [H]} 65 \textcolor{keywordtype}{real}, \textcolor{keywordtype}{dimension(:)}, \textcolor{keywordtype}{intent(in)} :: u\textcolor{comment}{ !< cell average properties (size N) [A]} 66 \textcolor{keywordtype}{real}, \textcolor{keywordtype}{dimension(:,:)}, \textcolor{keywordtype}{intent(inout)} :: edge\_values\textcolor{comment}{ !< Potentially modified edge values [A]} 67 \textcolor{keywordtype}{real}, \textcolor{keywordtype}{optional}, \textcolor{keywordtype}{intent(in)} :: h\_neglect\textcolor{comment}{ !< A negligibly small width [H]} 68 \textcolor{keywordtype}{logical}, \textcolor{keywordtype}{optional}, \textcolor{keywordtype}{intent(in)} :: answers\_2018\textcolor{comment}{ !< If true use older, less acccurate expressions.} 69 70 \textcolor{comment}{! Local variables} 71 \textcolor{keywordtype}{integer} :: k \textcolor{comment}{! Loop index} 72 \textcolor{keywordtype}{real} :: u\_l, u\_c, u\_r \textcolor{comment}{! Cell averages (left, center and right)} 73 \textcolor{keywordtype}{real} :: edge\_l, edge\_r \textcolor{comment}{! Edge values (left and right)} 74 \textcolor{keywordtype}{real} :: expr1, expr2 75 76 \textcolor{comment}{! Bound edge values} 77 \textcolor{keyword}{call }bound\_edge\_values( n, h, u, edge\_values, h\_neglect, answers\_2018 ) 78 79 \textcolor{comment}{! Make discontinuous edge values monotonic} 80 \textcolor{keyword}{call }check\_discontinuous\_edge\_values( n, u, edge\_values ) 81 82 \textcolor{comment}{! Loop on interior cells to apply the standard} 83 \textcolor{comment}{! PPM limiter (Colella & Woodward, JCP 84)} 84 \textcolor{keywordflow}{do} k = 2,n-1 85 86 \textcolor{comment}{! Get cell averages} 87 u\_l = u(k-1) 88 u\_c = u(k) 89 u\_r = u(k+1) 90 91 edge\_l = edge\_values(k,1) 92 edge\_r = edge\_values(k,2) 93 94 \textcolor{keywordflow}{if} ( (u\_r - u\_c)*(u\_c - u\_l) <= 0.0) \textcolor{keywordflow}{then} 95 \textcolor{comment}{! Flatten extremum} 96 edge\_l = u\_c 97 edge\_r = u\_c 98 \textcolor{keywordflow}{else} 99 expr1 = 3.0 * (edge\_r - edge\_l) * ( (u\_c - edge\_l) + (u\_c - edge\_r)) 100 expr2 = (edge\_r - edge\_l) * (edge\_r - edge\_l) 101 \textcolor{keywordflow}{if} ( expr1 > expr2 ) \textcolor{keywordflow}{then} 102 \textcolor{comment}{! Place extremum at right edge of cell by adjusting left edge value} 103 edge\_l = u\_c + 2.0 * ( u\_c - edge\_r ) 104 edge\_l = max( min( edge\_l, max(u\_l, u\_c) ), min(u\_l, u\_c) ) \textcolor{comment}{! In case of round off} 105 \textcolor{keywordflow}{elseif} ( expr1 < -expr2 ) \textcolor{keywordflow}{then} 106 \textcolor{comment}{! Place extremum at left edge of cell by adjusting right edge value} 107 edge\_r = u\_c + 2.0 * ( u\_c - edge\_l ) 108 edge\_r = max( min( edge\_r, max(u\_r, u\_c) ), min(u\_r, u\_c) ) \textcolor{comment}{! In case of round off} 109 \textcolor{keywordflow}{ endif} 110 \textcolor{keywordflow}{ endif} 111 \textcolor{comment}{! This checks that the difference in edge values is representable} 112 \textcolor{comment}{! and avoids overshoot problems due to round off.} 113 \textcolor{comment}{!### The 1.e-60 needs to have units of [A], so this dimensionally inconsisent.} 114 \textcolor{keywordflow}{if} ( abs( edge\_r - edge\_l )