\hypertarget{namespacep1m__functions}{}\section{p1m\+\_\+functions Module Reference} \label{namespacep1m__functions}\index{p1m\+\_\+functions@{p1m\+\_\+functions}} \subsection{Detailed Description} Linear interpolation functions. Date of creation\+: 2008.\+06.\+09 L. White. This module contains p1m (linear) interpolation routines. p1m interpolation is performed by estimating the edge values and linearly interpolating between them. \subsection*{Functions/\+Subroutines} \begin{DoxyCompactItemize} \item subroutine, public \hyperlink{namespacep1m__functions_a233a7aff25cf6581421f76bba053d758}{p1m\+\_\+interpolation} (N, h, u, edge\+\_\+values, ppoly\+\_\+coef, h\+\_\+neglect, answers\+\_\+2018) \begin{DoxyCompactList}\small\item\em Linearly interpolate between edge values. \end{DoxyCompactList}\item subroutine, public \hyperlink{namespacep1m__functions_a1a45cb8f3f794172d2a19b53e10416c6}{p1m\+\_\+boundary\+\_\+extrapolation} (N, h, u, edge\+\_\+values, ppoly\+\_\+coef) \begin{DoxyCompactList}\small\item\em Interpolation by linear polynomials within boundary cells. \end{DoxyCompactList}\end{DoxyCompactItemize} \subsection{Function/\+Subroutine Documentation} \mbox{\Hypertarget{namespacep1m__functions_a1a45cb8f3f794172d2a19b53e10416c6}\label{namespacep1m__functions_a1a45cb8f3f794172d2a19b53e10416c6}} \index{p1m\+\_\+functions@{p1m\+\_\+functions}!p1m\+\_\+boundary\+\_\+extrapolation@{p1m\+\_\+boundary\+\_\+extrapolation}} \index{p1m\+\_\+boundary\+\_\+extrapolation@{p1m\+\_\+boundary\+\_\+extrapolation}!p1m\+\_\+functions@{p1m\+\_\+functions}} \subsubsection{\texorpdfstring{p1m\+\_\+boundary\+\_\+extrapolation()}{p1m\_boundary\_extrapolation()}} {\footnotesize\ttfamily subroutine, public p1m\+\_\+functions\+::p1m\+\_\+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 }\end{DoxyParamCaption})} Interpolation 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. 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) \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 \end{DoxyParams} Definition at line 69 of file P1\+M\+\_\+functions.\+F90. \begin{DoxyCode} 69 \textcolor{comment}{! Arguments} 70 \textcolor{keywordtype}{integer}, \textcolor{keywordtype}{intent(in)} :: n\textcolor{comment}{ !< Number of cells} 71 \textcolor{keywordtype}{real}, \textcolor{keywordtype}{dimension(:)}, \textcolor{keywordtype}{intent(in)} :: h\textcolor{comment}{ !< cell widths (size N) [H]} 72 \textcolor{keywordtype}{real}, \textcolor{keywordtype}{dimension(:)}, \textcolor{keywordtype}{intent(in)} :: u\textcolor{comment}{ !< cell averages (size N) [A]} 73 \textcolor{keywordtype}{real}, \textcolor{keywordtype}{dimension(:,:)}, \textcolor{keywordtype}{intent(inout)} :: edge\_values\textcolor{comment}{ !< edge values of piecewise polynomials [A]} 74 \textcolor{keywordtype}{real}, \textcolor{keywordtype}{dimension(:,:)}, \textcolor{keywordtype}{intent(inout)} :: ppoly\_coef\textcolor{comment}{ !< coefficients of piecewise polynomials, mainly [A]} 75 76 \textcolor{comment}{! Local variables} 77 \textcolor{keywordtype}{real} :: u0, u1 \textcolor{comment}{! cell averages} 78 \textcolor{keywordtype}{real} :: h0, h1 \textcolor{comment}{! corresponding cell widths} 79 \textcolor{keywordtype}{real} :: slope \textcolor{comment}{! retained PLM slope} 80 \textcolor{keywordtype}{real} :: u0\_l, u0\_r \textcolor{comment}{! edge values} 81 82 \textcolor{comment}{! -----------------------------------------} 83 \textcolor{comment}{! Left edge value in the left boundary cell} 84 \textcolor{comment}{! -----------------------------------------} 85 h0 = h(1) 86 h1 = h(2) 87 88 u0 = u(1) 89 u1 = u(2) 90 91 \textcolor{comment}{! The standard PLM slope is computed as a first estimate for the} 92 \textcolor{comment}{! interpolation within the cell} 93 slope = 2.0 * ( u1 - u0 ) 94 95 \textcolor{comment}{! The right edge value is then computed and we check whether this} 96 \textcolor{comment}{! right edge value is consistent: it cannot be larger than the edge} 97 \textcolor{comment}{! value in the neighboring cell if the data set is increasing.} 98 \textcolor{comment}{! If the right value is found to too large, the slope is further limited} 99 \textcolor{comment}{! by using the edge value in the neighboring cell.} 100 u0\_r = u0 + 0.5 * slope 101 102 \textcolor{keywordflow}{if} ( (u1 - u0) * (edge\_values(2,1) - u0\_r) < 0.0 ) \textcolor{keywordflow}{then} 103 slope = 2.0 * ( edge\_values(2,1) - u0 ) 104 \textcolor{keywordflow}{ endif} 105 106 \textcolor{comment}{! Using the limited slope, the left edge value is reevaluated and} 107 \textcolor{comment}{! the interpolant coefficients recomputed} 108 \textcolor{keywordflow}{if} ( h0 /= 0.0 ) \textcolor{keywordflow}{then} 109 edge\_values(1,1) = u0 - 0.5 * slope 110 \textcolor{keywordflow}{else} 111 edge\_values(1,1) = u0 112 \textcolor{keywordflow}{ endif} 113 114 ppoly\_coef(1,1) = edge\_values(1,1) 115 ppoly\_coef(1,2) = edge\_values(1,2) - edge\_values(1,1) 116 117 \textcolor{comment}{! ------------------------------------------} 118 \textcolor{comment}{! Right edge value in the left boundary cell} 119 \textcolor{comment}{! ------------------------------------------} 120 h0 = h(n-1) 121 h1 = h(n) 122 123 u0 = u(n-1) 124 u1 = u(n) 125 126 slope = 2.0 * ( u1 - u0 ) 127 128 u0\_l = u1 - 0.5 * slope 129 130 \textcolor{keywordflow}{if} ( (u1 - u0) * (u0\_l - edge\_values(n-1,2)) < 0.0 ) \textcolor{keywordflow}{then} 131 slope = 2.0 * ( u1 - edge\_values(n-1,2) ) 132 \textcolor{keywordflow}{ endif} 133 134 \textcolor{keywordflow}{if} ( h1 /= 0.0 ) \textcolor{keywordflow}{then} 135 edge\_values(n,2) = u1 + 0.5 * slope 136 \textcolor{keywordflow}{else} 137 edge\_values(n,2) = u1 138 \textcolor{keywordflow}{ endif} 139 140 ppoly\_coef(n,1) = edge\_values(n,1) 141 ppoly\_coef(n,2) = edge\_values(n,2) - edge\_values(n,1) 142 \end{DoxyCode} \mbox{\Hypertarget{namespacep1m__functions_a233a7aff25cf6581421f76bba053d758}\label{namespacep1m__functions_a233a7aff25cf6581421f76bba053d758}} \index{p1m\+\_\+functions@{p1m\+\_\+functions}!p1m\+\_\+interpolation@{p1m\+\_\+interpolation}} \index{p1m\+\_\+interpolation@{p1m\+\_\+interpolation}!p1m\+\_\+functions@{p1m\+\_\+functions}} \subsubsection{\texorpdfstring{p1m\+\_\+interpolation()}{p1m\_interpolation()}} {\footnotesize\ttfamily subroutine, public p1m\+\_\+functions\+::p1m\+\_\+interpolation (\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, }\item[{logical, intent(in), optional}]{answers\+\_\+2018 }\end{DoxyParamCaption})} Linearly interpolate between edge values. The resulting piecewise interpolant is stored in \textquotesingle{}ppoly\textquotesingle{}. See \textquotesingle{}ppoly.\+F90\textquotesingle{} for a definition of this structure. The edge values M\+U\+ST have been estimated prior to calling this routine. The estimated edge values must be limited to ensure monotonicity of the interpolant. We also make sure that edge values are N\+OT discontinuous. 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) \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,out} & {\em ppoly\+\_\+coef} & Potentially modified piecewise polynomial coefficients, mainly \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 28 of file P1\+M\+\_\+functions.\+F90. \begin{DoxyCode} 28 \textcolor{keywordtype}{integer}, \textcolor{keywordtype}{intent(in)} :: n\textcolor{comment}{ !< Number of cells} 29 \textcolor{keywordtype}{real}, \textcolor{keywordtype}{dimension(:)}, \textcolor{keywordtype}{intent(in)} :: h\textcolor{comment}{ !< cell widths (size N) [H]} 30 \textcolor{keywordtype}{real}, \textcolor{keywordtype}{dimension(:)}, \textcolor{keywordtype}{intent(in)} :: u\textcolor{comment}{ !< cell average properties (size N) [A]} 31 \textcolor{keywordtype}{real}, \textcolor{keywordtype}{dimension(:,:)}, \textcolor{keywordtype}{intent(inout)} :: edge\_values\textcolor{comment}{ !< Potentially modified edge values [A]} 32 \textcolor{keywordtype}{real}, \textcolor{keywordtype}{dimension(:,:)}, \textcolor{keywordtype}{intent(inout)} :: ppoly\_coef\textcolor{comment}{ !< Potentially modified} 33 \textcolor{comment}{ !! piecewise polynomial coefficients, mainly [A]} 34 \textcolor{keywordtype}{real}, \textcolor{keywordtype}{optional}, \textcolor{keywordtype}{intent(in)} :: h\_neglect\textcolor{comment}{ !< A negligibly small width [H]} 35 \textcolor{keywordtype}{logical}, \textcolor{keywordtype}{optional}, \textcolor{keywordtype}{intent(in)} :: answers\_2018\textcolor{comment}{ !< If true use older, less acccurate expressions.} 36 37 \textcolor{comment}{! Local variables} 38 \textcolor{keywordtype}{integer} :: k \textcolor{comment}{! loop index} 39 \textcolor{keywordtype}{real} :: u0\_l, u0\_r \textcolor{comment}{! edge values (left and right)} 40 41 \textcolor{comment}{! Bound edge values (routine found in 'edge\_values.F90')} 42 \textcolor{keyword}{call }bound\_edge\_values( n, h, u, edge\_values, h\_neglect, answers\_2018 ) 43 44 \textcolor{comment}{! Systematically average discontinuous edge values (routine found in} 45 \textcolor{comment}{! 'edge\_values.F90')} 46 \textcolor{keyword}{call }average\_discontinuous\_edge\_values( n, edge\_values ) 47 48 \textcolor{comment}{! Loop on interior cells to build interpolants} 49 \textcolor{keywordflow}{do} k = 1,n 50 51 u0\_l = edge\_values(k,1) 52 u0\_r = edge\_values(k,2) 53 54 ppoly\_coef(k,1) = u0\_l 55 ppoly\_coef(k,2) = u0\_r - u0\_l 56 57 \textcolor{keywordflow}{ enddo} \textcolor{comment}{! end loop on interior cells} 58 \end{DoxyCode}