\hypertarget{namespacepqm__functions}{}\section{pqm\+\_\+functions Module Reference} \label{namespacepqm__functions}\index{pqm\_functions@{pqm\_functions}} \subsection{Detailed Description} Piecewise quartic reconstruction functions. Date of creation\+: 2008.\+06.\+06 L. White. This module contains routines that handle one-\/dimensionnal finite volume reconstruction using the piecewise quartic method (P\+QM). \subsection*{Functions/\+Subroutines} \begin{DoxyCompactItemize} \item subroutine, public \mbox{\hyperlink{namespacepqm__functions_afa6f7b5430011f03c428b329c5f42fae}{pqm\+\_\+reconstruction}} (N, h, u, edge\+\_\+values, edge\+\_\+slopes, ppoly\+\_\+coef, h\+\_\+neglect, answers\+\_\+2018) \begin{DoxyCompactList}\small\item\em Reconstruction by quartic polynomials within each cell. \end{DoxyCompactList}\item subroutine \mbox{\hyperlink{namespacepqm__functions_aab9f3108956943c39ed2d4b675d78021}{pqm\+\_\+limiter}} (N, h, u, edge\+\_\+values, edge\+\_\+slopes, h\+\_\+neglect, answers\+\_\+2018) \begin{DoxyCompactList}\small\item\em Limit the piecewise quartic method reconstruction. \end{DoxyCompactList}\item subroutine, public \mbox{\hyperlink{namespacepqm__functions_ad970064d6c3540fd00dff2453d0778d7}{pqm\+\_\+boundary\+\_\+extrapolation}} (N, h, u, edge\+\_\+values, ppoly\+\_\+coef) \begin{DoxyCompactList}\small\item\em Reconstruction by parabolas within boundary cells. \end{DoxyCompactList}\item subroutine, public \mbox{\hyperlink{namespacepqm__functions_a1e0e86d8470dd334b9cd676f511f6720}{pqm\+\_\+boundary\+\_\+extrapolation\+\_\+v1}} (N, h, u, edge\+\_\+values, edge\+\_\+slopes, ppoly\+\_\+coef, h\+\_\+neglect) \begin{DoxyCompactList}\small\item\em Reconstruction by parabolas within boundary cells. \end{DoxyCompactList}\end{DoxyCompactItemize} \subsection*{Variables} \begin{DoxyCompactItemize} \item \mbox{\Hypertarget{namespacepqm__functions_a3bd59442a660e0e66c29603c992932e6}\label{namespacepqm__functions_a3bd59442a660e0e66c29603c992932e6}} real, parameter \mbox{\hyperlink{namespacepqm__functions_a3bd59442a660e0e66c29603c992932e6}{hneglect\+\_\+dflt}} = 1.E-\/30 \begin{DoxyCompactList}\small\item\em Default negligible cell thickness. \end{DoxyCompactList}\end{DoxyCompactItemize} \subsection{Function/\+Subroutine Documentation} \mbox{\Hypertarget{namespacepqm__functions_ad970064d6c3540fd00dff2453d0778d7}\label{namespacepqm__functions_ad970064d6c3540fd00dff2453d0778d7}} \index{pqm\_functions@{pqm\_functions}!pqm\_boundary\_extrapolation@{pqm\_boundary\_extrapolation}} \index{pqm\_boundary\_extrapolation@{pqm\_boundary\_extrapolation}!pqm\_functions@{pqm\_functions}} \subsubsection{\texorpdfstring{pqm\_boundary\_extrapolation()}{pqm\_boundary\_extrapolation()}} {\footnotesize\ttfamily subroutine, public pqm\+\_\+functions\+::pqm\+\_\+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})} Reconstruction by parabolas within boundary cells. The following explanations apply to the left boundary cell. The same reasoning holds for the right boundary cell. A parabola needs to be built in the cell and requires three degrees of freedom, which are the right edge value and slope and the cell average. The right edge values and slopes are taken to be that of the neighboring cell (i.\+e., the left edge value and slope of the neighboring cell). The resulting parabola is not necessarily monotonic and the traditional P\+PM limiter is used to modify one of the edge values in order to yield a monotonic parabola. 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) \mbox{[}H\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em u} & cell averages (size N) \mbox{[}A\mbox{]} \\ \hline \mbox{\texttt{ in,out}} & {\em edge\+\_\+values} & Edge value of polynomial \mbox{[}A\mbox{]} \\ \hline \mbox{\texttt{ in,out}} & {\em ppoly\+\_\+coef} & Coefficients of polynomial, mainly \mbox{[}A\mbox{]} \\ \hline \end{DoxyParams} Definition at line 355 of file P\+Q\+M\+\_\+functions.\+F90. \begin{DoxyCode}{0} \DoxyCodeLine{355 \textcolor{keywordtype}{integer}, \textcolor{keywordtype}{intent(in)} :: N\textcolor{comment}{ !< Number of cells}} \DoxyCodeLine{356 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:)}, \textcolor{keywordtype}{intent(in)} :: h\textcolor{comment}{ !< cell widths (size N) [H]}} \DoxyCodeLine{357 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:)}, \textcolor{keywordtype}{intent(in)} :: u\textcolor{comment}{ !< cell averages (size N) [A]}} \DoxyCodeLine{358 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:,:)}, \textcolor{keywordtype}{intent(inout)} :: edge\_values\textcolor{comment}{ !< Edge value of polynomial [A]}} \DoxyCodeLine{359 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:,:)}, \textcolor{keywordtype}{intent(inout)} :: ppoly\_coef\textcolor{comment}{ !< Coefficients of polynomial, mainly [A]}} \DoxyCodeLine{360 \textcolor{comment}{! Local variables}} \DoxyCodeLine{361 \textcolor{keywordtype}{integer} :: i0, i1} \DoxyCodeLine{362 \textcolor{keywordtype}{ real} :: u0, u1} \DoxyCodeLine{363 \textcolor{keywordtype}{ real} :: h0, h1} \DoxyCodeLine{364 \textcolor{keywordtype}{ real} :: a, b, c, d, e} \DoxyCodeLine{365 \textcolor{keywordtype}{ real} :: u0\_l, u0\_r} \DoxyCodeLine{366 \textcolor{keywordtype}{ real} :: u1\_l, u1\_r} \DoxyCodeLine{367 \textcolor{keywordtype}{ real} :: slope} \DoxyCodeLine{368 \textcolor{keywordtype}{ real} :: exp1, exp2} \DoxyCodeLine{369 } \DoxyCodeLine{370 \textcolor{comment}{! ----- Left boundary -----}} \DoxyCodeLine{371 i0 = 1} \DoxyCodeLine{372 i1 = 2} \DoxyCodeLine{373 h0 = h(i0)} \DoxyCodeLine{374 h1 = h(i1)} \DoxyCodeLine{375 u0 = u(i0)} \DoxyCodeLine{376 u1 = u(i1)} \DoxyCodeLine{377 } \DoxyCodeLine{378 \textcolor{comment}{! Compute the left edge slope in neighboring cell and express it in}} \DoxyCodeLine{379 \textcolor{comment}{! the global coordinate system}} \DoxyCodeLine{380 b = ppoly\_coef(i1,2)} \DoxyCodeLine{381 u1\_r = b *(h0/h1) \textcolor{comment}{! derivative evaluated at xi = 0.0,}} \DoxyCodeLine{382 \textcolor{comment}{! expressed w.r.t. xi (local coord. system)}} \DoxyCodeLine{383 } \DoxyCodeLine{384 \textcolor{comment}{! Limit the right slope by the PLM limited slope}} \DoxyCodeLine{385 slope = 2.0 * ( u1 - u0 )} \DoxyCodeLine{386 \textcolor{keywordflow}{if} ( abs(u1\_r) > abs(slope) ) \textcolor{keywordflow}{then}} \DoxyCodeLine{387 u1\_r = slope} \DoxyCodeLine{388 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{389 } \DoxyCodeLine{390 \textcolor{comment}{! The right edge value in the boundary cell is taken to be the left}} \DoxyCodeLine{391 \textcolor{comment}{! edge value in the neighboring cell}} \DoxyCodeLine{392 u0\_r = edge\_values(i1,1)} \DoxyCodeLine{393 } \DoxyCodeLine{394 \textcolor{comment}{! Given the right edge value and slope, we determine the left}} \DoxyCodeLine{395 \textcolor{comment}{! edge value and slope by computing the parabola as determined by}} \DoxyCodeLine{396 \textcolor{comment}{! the right edge value and slope and the boundary cell average}} \DoxyCodeLine{397 u0\_l = 3.0 * u0 + 0.5 * u1\_r - 2.0 * u0\_r} \DoxyCodeLine{398 } \DoxyCodeLine{399 \textcolor{comment}{! Apply the traditional PPM limiter}} \DoxyCodeLine{400 exp1 = (u0\_r - u0\_l) * (u0 - 0.5*(u0\_l+u0\_r))} \DoxyCodeLine{401 exp2 = (u0\_r - u0\_l) * (u0\_r - u0\_l) / 6.0} \DoxyCodeLine{402 } \DoxyCodeLine{403 \textcolor{keywordflow}{if} ( exp1 > exp2 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{404 u0\_l = 3.0 * u0 - 2.0 * u0\_r} \DoxyCodeLine{405 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{406 } \DoxyCodeLine{407 \textcolor{keywordflow}{if} ( exp1 < -exp2 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{408 u0\_r = 3.0 * u0 - 2.0 * u0\_l} \DoxyCodeLine{409 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{410 } \DoxyCodeLine{411 edge\_values(i0,1) = u0\_l} \DoxyCodeLine{412 edge\_values(i0,2) = u0\_r} \DoxyCodeLine{413 } \DoxyCodeLine{414 a = u0\_l} \DoxyCodeLine{415 b = 6.0 * u0 - 4.0 * u0\_l - 2.0 * u0\_r} \DoxyCodeLine{416 c = 3.0 * ( u0\_r + u0\_l - 2.0 * u0 )} \DoxyCodeLine{417 } \DoxyCodeLine{418 \textcolor{comment}{! The quartic is reduced to a parabola in the boundary cell}} \DoxyCodeLine{419 ppoly\_coef(i0,1) = a} \DoxyCodeLine{420 ppoly\_coef(i0,2) = b} \DoxyCodeLine{421 ppoly\_coef(i0,3) = c} \DoxyCodeLine{422 ppoly\_coef(i0,4) = 0.0} \DoxyCodeLine{423 ppoly\_coef(i0,5) = 0.0} \DoxyCodeLine{424 } \DoxyCodeLine{425 \textcolor{comment}{! ----- Right boundary -----}} \DoxyCodeLine{426 i0 = n-1} \DoxyCodeLine{427 i1 = n} \DoxyCodeLine{428 h0 = h(i0)} \DoxyCodeLine{429 h1 = h(i1)} \DoxyCodeLine{430 u0 = u(i0)} \DoxyCodeLine{431 u1 = u(i1)} \DoxyCodeLine{432 } \DoxyCodeLine{433 \textcolor{comment}{! Compute the right edge slope in neighboring cell and express it in}} \DoxyCodeLine{434 \textcolor{comment}{! the global coordinate system}} \DoxyCodeLine{435 b = ppoly\_coef(i0,2)} \DoxyCodeLine{436 c = ppoly\_coef(i0,3)} \DoxyCodeLine{437 d = ppoly\_coef(i0,4)} \DoxyCodeLine{438 e = ppoly\_coef(i0,5)} \DoxyCodeLine{439 u1\_l = (b + 2*c + 3*d + 4*e) \textcolor{comment}{! derivative evaluated at xi = 1.0}} \DoxyCodeLine{440 u1\_l = u1\_l * (h1/h0)} \DoxyCodeLine{441 } \DoxyCodeLine{442 \textcolor{comment}{! Limit the left slope by the PLM limited slope}} \DoxyCodeLine{443 slope = 2.0 * ( u1 - u0 )} \DoxyCodeLine{444 \textcolor{keywordflow}{if} ( abs(u1\_l) > abs(slope) ) \textcolor{keywordflow}{then}} \DoxyCodeLine{445 u1\_l = slope} \DoxyCodeLine{446 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{447 } \DoxyCodeLine{448 \textcolor{comment}{! The left edge value in the boundary cell is taken to be the right}} \DoxyCodeLine{449 \textcolor{comment}{! edge value in the neighboring cell}} \DoxyCodeLine{450 u0\_l = edge\_values(i0,2)} \DoxyCodeLine{451 } \DoxyCodeLine{452 \textcolor{comment}{! Given the left edge value and slope, we determine the right}} \DoxyCodeLine{453 \textcolor{comment}{! edge value and slope by computing the parabola as determined by}} \DoxyCodeLine{454 \textcolor{comment}{! the left edge value and slope and the boundary cell average}} \DoxyCodeLine{455 u0\_r = 3.0 * u1 - 0.5 * u1\_l - 2.0 * u0\_l} \DoxyCodeLine{456 } \DoxyCodeLine{457 \textcolor{comment}{! Apply the traditional PPM limiter}} \DoxyCodeLine{458 exp1 = (u0\_r - u0\_l) * (u1 - 0.5*(u0\_l+u0\_r))} \DoxyCodeLine{459 exp2 = (u0\_r - u0\_l) * (u0\_r - u0\_l) / 6.0} \DoxyCodeLine{460 } \DoxyCodeLine{461 \textcolor{keywordflow}{if} ( exp1 > exp2 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{462 u0\_l = 3.0 * u1 - 2.0 * u0\_r} \DoxyCodeLine{463 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{464 } \DoxyCodeLine{465 \textcolor{keywordflow}{if} ( exp1 < -exp2 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{466 u0\_r = 3.0 * u1 - 2.0 * u0\_l} \DoxyCodeLine{467 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{468 } \DoxyCodeLine{469 edge\_values(i1,1) = u0\_l} \DoxyCodeLine{470 edge\_values(i1,2) = u0\_r} \DoxyCodeLine{471 } \DoxyCodeLine{472 a = u0\_l} \DoxyCodeLine{473 b = 6.0 * u1 - 4.0 * u0\_l - 2.0 * u0\_r} \DoxyCodeLine{474 c = 3.0 * ( u0\_r + u0\_l - 2.0 * u1 )} \DoxyCodeLine{475 } \DoxyCodeLine{476 \textcolor{comment}{! The quartic is reduced to a parabola in the boundary cell}} \DoxyCodeLine{477 ppoly\_coef(i1,1) = a} \DoxyCodeLine{478 ppoly\_coef(i1,2) = b} \DoxyCodeLine{479 ppoly\_coef(i1,3) = c} \DoxyCodeLine{480 ppoly\_coef(i1,4) = 0.0} \DoxyCodeLine{481 ppoly\_coef(i1,5) = 0.0} \DoxyCodeLine{482 } \end{DoxyCode} \mbox{\Hypertarget{namespacepqm__functions_a1e0e86d8470dd334b9cd676f511f6720}\label{namespacepqm__functions_a1e0e86d8470dd334b9cd676f511f6720}} \index{pqm\_functions@{pqm\_functions}!pqm\_boundary\_extrapolation\_v1@{pqm\_boundary\_extrapolation\_v1}} \index{pqm\_boundary\_extrapolation\_v1@{pqm\_boundary\_extrapolation\_v1}!pqm\_functions@{pqm\_functions}} \subsubsection{\texorpdfstring{pqm\_boundary\_extrapolation\_v1()}{pqm\_boundary\_extrapolation\_v1()}} {\footnotesize\ttfamily subroutine, public pqm\+\_\+functions\+::pqm\+\_\+boundary\+\_\+extrapolation\+\_\+v1 (\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)}]{edge\+\_\+slopes, }\item[{real, dimension(\+:,\+:), intent(inout)}]{ppoly\+\_\+coef, }\item[{real, intent(in), optional}]{h\+\_\+neglect }\end{DoxyParamCaption})} Reconstruction by parabolas within boundary cells. The following explanations apply to the left boundary cell. The same reasoning holds for the right boundary cell. A parabola needs to be built in the cell and requires three degrees of freedom, which are the right edge value and slope and the cell average. The right edge values and slopes are taken to be that of the neighboring cell (i.\+e., the left edge value and slope of the neighboring cell). The resulting parabola is not necessarily monotonic and the traditional P\+PM limiter is used to modify one of the edge values in order to yield a monotonic parabola. 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) \mbox{[}H\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em u} & cell averages (size N) \mbox{[}A\mbox{]} \\ \hline \mbox{\texttt{ in,out}} & {\em edge\+\_\+values} & Edge value of polynomial \mbox{[}A\mbox{]} \\ \hline \mbox{\texttt{ in,out}} & {\em edge\+\_\+slopes} & Edge slope of polynomial \mbox{[}A H-\/1\mbox{]} \\ \hline \mbox{\texttt{ in,out}} & {\em ppoly\+\_\+coef} & Coefficients of polynomial, mainly \mbox{[}A\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em h\+\_\+neglect} & A negligibly small width for the purpose of cell reconstructions \mbox{[}H\mbox{]} \\ \hline \end{DoxyParams} Definition at line 502 of file P\+Q\+M\+\_\+functions.\+F90. \begin{DoxyCode}{0} \DoxyCodeLine{502 \textcolor{keywordtype}{integer}, \textcolor{keywordtype}{intent(in)} :: N\textcolor{comment}{ !< Number of cells}} \DoxyCodeLine{503 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:)}, \textcolor{keywordtype}{intent(in)} :: h\textcolor{comment}{ !< cell widths (size N) [H]}} \DoxyCodeLine{504 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:)}, \textcolor{keywordtype}{intent(in)} :: u\textcolor{comment}{ !< cell averages (size N) [A]}} \DoxyCodeLine{505 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:,:)}, \textcolor{keywordtype}{intent(inout)} :: edge\_values\textcolor{comment}{ !< Edge value of polynomial [A]}} \DoxyCodeLine{506 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:,:)}, \textcolor{keywordtype}{intent(inout)} :: edge\_slopes\textcolor{comment}{ !< Edge slope of polynomial [A H-1]}} \DoxyCodeLine{507 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:,:)}, \textcolor{keywordtype}{intent(inout)} :: ppoly\_coef\textcolor{comment}{ !< Coefficients of polynomial, mainly [A]}} \DoxyCodeLine{508 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{optional}, \textcolor{keywordtype}{intent(in)} :: h\_neglect\textcolor{comment}{ !< A negligibly small width for}} \DoxyCodeLine{509 \textcolor{comment}{ !! the purpose of cell reconstructions [H]}} \DoxyCodeLine{510 \textcolor{comment}{! Local variables}} \DoxyCodeLine{511 \textcolor{keywordtype}{integer} :: i0, i1} \DoxyCodeLine{512 \textcolor{keywordtype}{integer} :: inflexion\_l} \DoxyCodeLine{513 \textcolor{keywordtype}{integer} :: inflexion\_r} \DoxyCodeLine{514 \textcolor{keywordtype}{ real} :: u0, u1, um} \DoxyCodeLine{515 \textcolor{keywordtype}{ real} :: h0, h1} \DoxyCodeLine{516 \textcolor{keywordtype}{ real} :: a, b, c, d, e} \DoxyCodeLine{517 \textcolor{keywordtype}{ real} :: ar, br, beta} \DoxyCodeLine{518 \textcolor{keywordtype}{ real} :: u0\_l, u0\_r} \DoxyCodeLine{519 \textcolor{keywordtype}{ real} :: u1\_l, u1\_r} \DoxyCodeLine{520 \textcolor{keywordtype}{ real} :: u\_plm} \DoxyCodeLine{521 \textcolor{keywordtype}{ real} :: slope} \DoxyCodeLine{522 \textcolor{keywordtype}{ real} :: alpha1, alpha2, alpha3} \DoxyCodeLine{523 \textcolor{keywordtype}{ real} :: rho, sqrt\_rho} \DoxyCodeLine{524 \textcolor{keywordtype}{ real} :: gradient1, gradient2} \DoxyCodeLine{525 \textcolor{keywordtype}{ real} :: x1, x2} \DoxyCodeLine{526 \textcolor{keywordtype}{ real} :: hNeglect} \DoxyCodeLine{527 } \DoxyCodeLine{528 hneglect = hneglect\_dflt ; \textcolor{keywordflow}{if} (\textcolor{keyword}{present}(h\_neglect)) hneglect = h\_neglect} \DoxyCodeLine{529 } \DoxyCodeLine{530 \textcolor{comment}{! ----- Left boundary (TOP) -----}} \DoxyCodeLine{531 i0 = 1} \DoxyCodeLine{532 i1 = 2} \DoxyCodeLine{533 h0 = h(i0)} \DoxyCodeLine{534 h1 = h(i1)} \DoxyCodeLine{535 u0 = u(i0)} \DoxyCodeLine{536 u1 = u(i1)} \DoxyCodeLine{537 um = u0} \DoxyCodeLine{538 } \DoxyCodeLine{539 \textcolor{comment}{! Compute real slope and express it w.r.t. local coordinate system}} \DoxyCodeLine{540 \textcolor{comment}{! within boundary cell}} \DoxyCodeLine{541 slope = 2.0 * ( u1 - u0 ) / ( ( h0 + h1 ) + hneglect )} \DoxyCodeLine{542 slope = slope * h0} \DoxyCodeLine{543 } \DoxyCodeLine{544 \textcolor{comment}{! The right edge value and slope of the boundary cell are taken to be the}} \DoxyCodeLine{545 \textcolor{comment}{! left edge value and slope of the adjacent cell}} \DoxyCodeLine{546 a = ppoly\_coef(i1,1)} \DoxyCodeLine{547 b = ppoly\_coef(i1,2)} \DoxyCodeLine{548 } \DoxyCodeLine{549 u0\_r = a \textcolor{comment}{! edge value}} \DoxyCodeLine{550 u1\_r = b / (h1 + hneglect) \textcolor{comment}{! edge slope (w.r.t. global coord.)}} \DoxyCodeLine{551 } \DoxyCodeLine{552 \textcolor{comment}{! Compute coefficient for rational function based on mean and right}} \DoxyCodeLine{553 \textcolor{comment}{! edge value and slope}} \DoxyCodeLine{554 \textcolor{keywordflow}{if} (u1\_r /= 0.) \textcolor{keywordflow}{then} \textcolor{comment}{! HACK by AJA}} \DoxyCodeLine{555 beta = 2.0 * ( u0\_r - um ) / ( (h0 + hneglect)*u1\_r) - 1.0} \DoxyCodeLine{556 \textcolor{keywordflow}{else}} \DoxyCodeLine{557 beta = 0.} \DoxyCodeLine{558 \textcolor{keywordflow}{ endif} \textcolor{comment}{! HACK by AJA}} \DoxyCodeLine{559 br = u0\_r + beta*u0\_r - um} \DoxyCodeLine{560 ar = um + beta*um - br} \DoxyCodeLine{561 } \DoxyCodeLine{562 \textcolor{comment}{! Left edge value estimate based on rational function}} \DoxyCodeLine{563 u0\_l = ar} \DoxyCodeLine{564 } \DoxyCodeLine{565 \textcolor{comment}{! Edge value estimate based on PLM}} \DoxyCodeLine{566 u\_plm = um - 0.5 * slope} \DoxyCodeLine{567 } \DoxyCodeLine{568 \textcolor{comment}{! Check whether the left edge value is bounded by the mean and}} \DoxyCodeLine{569 \textcolor{comment}{! the PLM edge value. If so, keep it and compute left edge slope}} \DoxyCodeLine{570 \textcolor{comment}{! based on the rational function. If not, keep the PLM edge value and}} \DoxyCodeLine{571 \textcolor{comment}{! compute corresponding slope.}} \DoxyCodeLine{572 \textcolor{keywordflow}{if} ( abs(um-u0\_l) < abs(um-u\_plm) ) \textcolor{keywordflow}{then}} \DoxyCodeLine{573 u1\_l = 2.0 * ( br - ar*beta)} \DoxyCodeLine{574 u1\_l = u1\_l / (h0 + hneglect)} \DoxyCodeLine{575 \textcolor{keywordflow}{else}} \DoxyCodeLine{576 u0\_l = u\_plm} \DoxyCodeLine{577 u1\_l = slope / (h0 + hneglect)} \DoxyCodeLine{578 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{579 } \DoxyCodeLine{580 \textcolor{comment}{! Monotonize quartic}} \DoxyCodeLine{581 inflexion\_l = 0} \DoxyCodeLine{582 } \DoxyCodeLine{583 a = u0\_l} \DoxyCodeLine{584 b = h0 * u1\_l} \DoxyCodeLine{585 c = 30.0 * um - 12.0*u0\_r - 18.0*u0\_l + 1.5*h0*(u1\_r - 3.0*u1\_l)} \DoxyCodeLine{586 d = -60.0 * um + h0 *(6.0*u1\_l - 4.0*u1\_r) + 28.0*u0\_r + 32.0*u0\_l} \DoxyCodeLine{587 e = 30.0 * um + 2.5*h0*(u1\_r - u1\_l) - 15.0*(u0\_l + u0\_r)} \DoxyCodeLine{588 } \DoxyCodeLine{589 alpha1 = 6*e} \DoxyCodeLine{590 alpha2 = 3*d} \DoxyCodeLine{591 alpha3 = c} \DoxyCodeLine{592 } \DoxyCodeLine{593 rho = alpha2 * alpha2 - 4.0 * alpha1 * alpha3} \DoxyCodeLine{594 } \DoxyCodeLine{595 \textcolor{comment}{! Check whether inflexion points exist. If so, transform the quartic}} \DoxyCodeLine{596 \textcolor{comment}{! so that both inflexion points coalesce on the left edge.}} \DoxyCodeLine{597 \textcolor{keywordflow}{if} (( alpha1 /= 0.0 ) .and. ( rho >= 0.0 )) \textcolor{keywordflow}{then}} \DoxyCodeLine{598 } \DoxyCodeLine{599 sqrt\_rho = sqrt( rho )} \DoxyCodeLine{600 } \DoxyCodeLine{601 x1 = 0.5 * ( - alpha2 - sqrt\_rho ) / alpha1} \DoxyCodeLine{602 \textcolor{keywordflow}{if} ( (x1 > 0.0) .and. (x1 < 1.0) ) \textcolor{keywordflow}{then}} \DoxyCodeLine{603 gradient1 = 4.0 * e * (x1**3) + 3.0 * d * (x1**2) + 2.0 * c * x1 + b} \DoxyCodeLine{604 \textcolor{keywordflow}{if} ( gradient1 * slope < 0.0 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{605 inflexion\_l = 1} \DoxyCodeLine{606 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{607 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{608 } \DoxyCodeLine{609 x2 = 0.5 * ( - alpha2 + sqrt\_rho ) / alpha1} \DoxyCodeLine{610 \textcolor{keywordflow}{if} ( (x2 > 0.0) .and. (x2 < 1.0) ) \textcolor{keywordflow}{then}} \DoxyCodeLine{611 gradient2 = 4.0 * e * (x2**3) + 3.0 * d * (x2**2) + 2.0 * c * x2 + b} \DoxyCodeLine{612 \textcolor{keywordflow}{if} ( gradient2 * slope < 0.0 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{613 inflexion\_l = 1} \DoxyCodeLine{614 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{615 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{616 } \DoxyCodeLine{617 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{618 } \DoxyCodeLine{619 \textcolor{keywordflow}{if} (( alpha1 == 0.0 ) .and. ( alpha2 /= 0.0 )) \textcolor{keywordflow}{then}} \DoxyCodeLine{620 } \DoxyCodeLine{621 x1 = - alpha3 / alpha2} \DoxyCodeLine{622 \textcolor{keywordflow}{if} ( (x1 >= 0.0) .and. (x1 <= 1.0) ) \textcolor{keywordflow}{then}} \DoxyCodeLine{623 gradient1 = 3.0 * d * (x1**2) + 2.0 * c * x1 + b} \DoxyCodeLine{624 \textcolor{keywordflow}{if} ( gradient1 * slope < 0.0 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{625 inflexion\_l = 1} \DoxyCodeLine{626 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{627 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{628 } \DoxyCodeLine{629 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{630 } \DoxyCodeLine{631 \textcolor{keywordflow}{if} ( inflexion\_l == 1 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{632 } \DoxyCodeLine{633 \textcolor{comment}{! We modify the edge slopes so that both inflexion points}} \DoxyCodeLine{634 \textcolor{comment}{! collapse onto the left edge}} \DoxyCodeLine{635 u1\_l = ( 10.0 * um - 2.0 * u0\_r - 8.0 * u0\_l ) / (3.0*h0 + hneglect)} \DoxyCodeLine{636 u1\_r = ( -10.0 * um + 6.0 * u0\_r + 4.0 * u0\_l ) / (h0 + hneglect)} \DoxyCodeLine{637 } \DoxyCodeLine{638 \textcolor{comment}{! One of the modified slopes might be inconsistent. When that happens,}} \DoxyCodeLine{639 \textcolor{comment}{! the inconsistent slope is set equal to zero and the opposite edge value}} \DoxyCodeLine{640 \textcolor{comment}{! and edge slope are modified in compliance with the fact that both}} \DoxyCodeLine{641 \textcolor{comment}{! inflexion points must still be located on the left edge}} \DoxyCodeLine{642 \textcolor{keywordflow}{if} ( u1\_l * slope < 0.0 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{643 } \DoxyCodeLine{644 u1\_l = 0.0} \DoxyCodeLine{645 u0\_r = 5.0 * um - 4.0 * u0\_l} \DoxyCodeLine{646 u1\_r = 20.0 * (um - u0\_l) / ( h0 + hneglect )} \DoxyCodeLine{647 } \DoxyCodeLine{648 \textcolor{keywordflow}{elseif} ( u1\_r * slope < 0.0 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{649 } \DoxyCodeLine{650 u1\_r = 0.0} \DoxyCodeLine{651 u0\_l = (5.0*um - 3.0*u0\_r) / 2.0} \DoxyCodeLine{652 u1\_l = 10.0 * (-um + u0\_r) / (3.0 * h0 + hneglect )} \DoxyCodeLine{653 } \DoxyCodeLine{654 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{655 } \DoxyCodeLine{656 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{657 } \DoxyCodeLine{658 \textcolor{comment}{! Store edge values, edge slopes and coefficients}} \DoxyCodeLine{659 edge\_values(i0,1) = u0\_l} \DoxyCodeLine{660 edge\_values(i0,2) = u0\_r} \DoxyCodeLine{661 edge\_slopes(i0,1) = u1\_l} \DoxyCodeLine{662 edge\_slopes(i0,2) = u1\_r} \DoxyCodeLine{663 } \DoxyCodeLine{664 a = u0\_l} \DoxyCodeLine{665 b = h0 * u1\_l} \DoxyCodeLine{666 c = 30.0 * um - 12.0*u0\_r - 18.0*u0\_l + 1.5*h0*(u1\_r - 3.0*u1\_l)} \DoxyCodeLine{667 d = -60.0 * um + h0 *(6.0*u1\_l - 4.0*u1\_r) + 28.0*u0\_r + 32.0*u0\_l} \DoxyCodeLine{668 e = 30.0 * um + 2.5*h0*(u1\_r - u1\_l) - 15.0*(u0\_l + u0\_r)} \DoxyCodeLine{669 } \DoxyCodeLine{670 \textcolor{comment}{! Store coefficients}} \DoxyCodeLine{671 ppoly\_coef(i0,1) = a} \DoxyCodeLine{672 ppoly\_coef(i0,2) = b} \DoxyCodeLine{673 ppoly\_coef(i0,3) = c} \DoxyCodeLine{674 ppoly\_coef(i0,4) = d} \DoxyCodeLine{675 ppoly\_coef(i0,5) = e} \DoxyCodeLine{676 } \DoxyCodeLine{677 \textcolor{comment}{! ----- Right boundary (BOTTOM) -----}} \DoxyCodeLine{678 i0 = n-1} \DoxyCodeLine{679 i1 = n} \DoxyCodeLine{680 h0 = h(i0)} \DoxyCodeLine{681 h1 = h(i1)} \DoxyCodeLine{682 u0 = u(i0)} \DoxyCodeLine{683 u1 = u(i1)} \DoxyCodeLine{684 um = u1} \DoxyCodeLine{685 } \DoxyCodeLine{686 \textcolor{comment}{! Compute real slope and express it w.r.t. local coordinate system}} \DoxyCodeLine{687 \textcolor{comment}{! within boundary cell}} \DoxyCodeLine{688 slope = 2.0 * ( u1 - u0 ) / ( h0 + h1 )} \DoxyCodeLine{689 slope = slope * h1} \DoxyCodeLine{690 } \DoxyCodeLine{691 \textcolor{comment}{! The left edge value and slope of the boundary cell are taken to be the}} \DoxyCodeLine{692 \textcolor{comment}{! right edge value and slope of the adjacent cell}} \DoxyCodeLine{693 a = ppoly\_coef(i0,1)} \DoxyCodeLine{694 b = ppoly\_coef(i0,2)} \DoxyCodeLine{695 c = ppoly\_coef(i0,3)} \DoxyCodeLine{696 d = ppoly\_coef(i0,4)} \DoxyCodeLine{697 e = ppoly\_coef(i0,5)} \DoxyCodeLine{698 u0\_l = a + b + c + d + e \textcolor{comment}{! edge value}} \DoxyCodeLine{699 u1\_l = (b + 2*c + 3*d + 4*e) / h0 \textcolor{comment}{! edge slope (w.r.t. global coord.)}} \DoxyCodeLine{700 } \DoxyCodeLine{701 \textcolor{comment}{! Compute coefficient for rational function based on mean and left}} \DoxyCodeLine{702 \textcolor{comment}{! edge value and slope}} \DoxyCodeLine{703 \textcolor{keywordflow}{if} (um-u0\_l /= 0.) \textcolor{keywordflow}{then} \textcolor{comment}{! HACK by AJA}} \DoxyCodeLine{704 beta = 0.5*h1*u1\_l / (um-u0\_l) - 1.0} \DoxyCodeLine{705 \textcolor{keywordflow}{else}} \DoxyCodeLine{706 beta = 0.} \DoxyCodeLine{707 \textcolor{keywordflow}{ endif} \textcolor{comment}{! HACK by AJA}} \DoxyCodeLine{708 br = beta*um + um - u0\_l} \DoxyCodeLine{709 ar = u0\_l} \DoxyCodeLine{710 } \DoxyCodeLine{711 \textcolor{comment}{! Right edge value estimate based on rational function}} \DoxyCodeLine{712 \textcolor{keywordflow}{if} (1+beta /= 0.) \textcolor{keywordflow}{then} \textcolor{comment}{! HACK by AJA}} \DoxyCodeLine{713 u0\_r = (ar + 2*br + beta*br ) / ((1+beta)*(1+beta))} \DoxyCodeLine{714 \textcolor{keywordflow}{else}} \DoxyCodeLine{715 u0\_r = um + 0.5 * slope \textcolor{comment}{! PLM}} \DoxyCodeLine{716 \textcolor{keywordflow}{ endif} \textcolor{comment}{! HACK by AJA}} \DoxyCodeLine{717 } \DoxyCodeLine{718 \textcolor{comment}{! Right edge value estimate based on PLM}} \DoxyCodeLine{719 u\_plm = um + 0.5 * slope} \DoxyCodeLine{720 } \DoxyCodeLine{721 \textcolor{comment}{! Check whether the right edge value is bounded by the mean and}} \DoxyCodeLine{722 \textcolor{comment}{! the PLM edge value. If so, keep it and compute right edge slope}} \DoxyCodeLine{723 \textcolor{comment}{! based on the rational function. If not, keep the PLM edge value and}} \DoxyCodeLine{724 \textcolor{comment}{! compute corresponding slope.}} \DoxyCodeLine{725 \textcolor{keywordflow}{if} ( abs(um-u0\_r) < abs(um-u\_plm) ) \textcolor{keywordflow}{then}} \DoxyCodeLine{726 u1\_r = 2.0 * ( br - ar*beta ) / ( (1+beta)*(1+beta)*(1+beta) )} \DoxyCodeLine{727 u1\_r = u1\_r / h1} \DoxyCodeLine{728 \textcolor{keywordflow}{else}} \DoxyCodeLine{729 u0\_r = u\_plm} \DoxyCodeLine{730 u1\_r = slope / h1} \DoxyCodeLine{731 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{732 } \DoxyCodeLine{733 \textcolor{comment}{! Monotonize quartic}} \DoxyCodeLine{734 inflexion\_r = 0} \DoxyCodeLine{735 } \DoxyCodeLine{736 a = u0\_l} \DoxyCodeLine{737 b = h1 * u1\_l} \DoxyCodeLine{738 c = 30.0 * um - 12.0*u0\_r - 18.0*u0\_l + 1.5*h1*(u1\_r - 3.0*u1\_l)} \DoxyCodeLine{739 d = -60.0 * um + h1*(6.0*u1\_l - 4.0*u1\_r) + 28.0*u0\_r + 32.0*u0\_l} \DoxyCodeLine{740 e = 30.0 * um + 2.5*h1*(u1\_r - u1\_l) - 15.0*(u0\_l + u0\_r)} \DoxyCodeLine{741 } \DoxyCodeLine{742 alpha1 = 6*e} \DoxyCodeLine{743 alpha2 = 3*d} \DoxyCodeLine{744 alpha3 = c} \DoxyCodeLine{745 } \DoxyCodeLine{746 rho = alpha2 * alpha2 - 4.0 * alpha1 * alpha3} \DoxyCodeLine{747 } \DoxyCodeLine{748 \textcolor{comment}{! Check whether inflexion points exist. If so, transform the quartic}} \DoxyCodeLine{749 \textcolor{comment}{! so that both inflexion points coalesce on the right edge.}} \DoxyCodeLine{750 \textcolor{keywordflow}{if} (( alpha1 /= 0.0 ) .and. ( rho >= 0.0 )) \textcolor{keywordflow}{then}} \DoxyCodeLine{751 } \DoxyCodeLine{752 sqrt\_rho = sqrt( rho )} \DoxyCodeLine{753 } \DoxyCodeLine{754 x1 = 0.5 * ( - alpha2 - sqrt\_rho ) / alpha1} \DoxyCodeLine{755 \textcolor{keywordflow}{if} ( (x1 > 0.0) .and. (x1 < 1.0) ) \textcolor{keywordflow}{then}} \DoxyCodeLine{756 gradient1 = 4.0 * e * (x1**3) + 3.0 * d * (x1**2) + 2.0 * c * x1 + b} \DoxyCodeLine{757 \textcolor{keywordflow}{if} ( gradient1 * slope < 0.0 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{758 inflexion\_r = 1} \DoxyCodeLine{759 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{760 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{761 } \DoxyCodeLine{762 x2 = 0.5 * ( - alpha2 + sqrt\_rho ) / alpha1} \DoxyCodeLine{763 \textcolor{keywordflow}{if} ( (x2 > 0.0) .and. (x2 < 1.0) ) \textcolor{keywordflow}{then}} \DoxyCodeLine{764 gradient2 = 4.0 * e * (x2**3) + 3.0 * d * (x2**2) + 2.0 * c * x2 + b} \DoxyCodeLine{765 \textcolor{keywordflow}{if} ( gradient2 * slope < 0.0 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{766 inflexion\_r = 1} \DoxyCodeLine{767 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{768 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{769 } \DoxyCodeLine{770 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{771 } \DoxyCodeLine{772 \textcolor{keywordflow}{if} (( alpha1 == 0.0 ) .and. ( alpha2 /= 0.0 )) \textcolor{keywordflow}{then}} \DoxyCodeLine{773 } \DoxyCodeLine{774 x1 = - alpha3 / alpha2} \DoxyCodeLine{775 \textcolor{keywordflow}{if} ( (x1 >= 0.0) .and. (x1 <= 1.0) ) \textcolor{keywordflow}{then}} \DoxyCodeLine{776 gradient1 = 3.0 * d * (x1**2) + 2.0 * c * x1 + b} \DoxyCodeLine{777 \textcolor{keywordflow}{if} ( gradient1 * slope < 0.0 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{778 inflexion\_r = 1} \DoxyCodeLine{779 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{780 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{781 } \DoxyCodeLine{782 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{783 } \DoxyCodeLine{784 \textcolor{keywordflow}{if} ( inflexion\_r == 1 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{785 } \DoxyCodeLine{786 \textcolor{comment}{! We modify the edge slopes so that both inflexion points}} \DoxyCodeLine{787 \textcolor{comment}{! collapse onto the right edge}} \DoxyCodeLine{788 u1\_r = ( -10.0 * um + 8.0 * u0\_r + 2.0 * u0\_l ) / (3.0 * h1)} \DoxyCodeLine{789 u1\_l = ( 10.0 * um - 4.0 * u0\_r - 6.0 * u0\_l ) / h1} \DoxyCodeLine{790 } \DoxyCodeLine{791 \textcolor{comment}{! One of the modified slopes might be inconsistent. When that happens,}} \DoxyCodeLine{792 \textcolor{comment}{! the inconsistent slope is set equal to zero and the opposite edge value}} \DoxyCodeLine{793 \textcolor{comment}{! and edge slope are modified in compliance with the fact that both}} \DoxyCodeLine{794 \textcolor{comment}{! inflexion points must still be located on the right edge}} \DoxyCodeLine{795 \textcolor{keywordflow}{if} ( u1\_l * slope < 0.0 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{796 } \DoxyCodeLine{797 u1\_l = 0.0} \DoxyCodeLine{798 u0\_r = ( 5.0 * um - 3.0 * u0\_l ) / 2.0} \DoxyCodeLine{799 u1\_r = 10.0 * (um - u0\_l) / (3.0 * h1)} \DoxyCodeLine{800 } \DoxyCodeLine{801 \textcolor{keywordflow}{elseif} ( u1\_r * slope < 0.0 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{802 } \DoxyCodeLine{803 u1\_r = 0.0} \DoxyCodeLine{804 u0\_l = 5.0 * um - 4.0 * u0\_r} \DoxyCodeLine{805 u1\_l = 20.0 * ( -um + u0\_r ) / h1} \DoxyCodeLine{806 } \DoxyCodeLine{807 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{808 } \DoxyCodeLine{809 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{810 } \DoxyCodeLine{811 \textcolor{comment}{! Store edge values, edge slopes and coefficients}} \DoxyCodeLine{812 edge\_values(i1,1) = u0\_l} \DoxyCodeLine{813 edge\_values(i1,2) = u0\_r} \DoxyCodeLine{814 edge\_slopes(i1,1) = u1\_l} \DoxyCodeLine{815 edge\_slopes(i1,2) = u1\_r} \DoxyCodeLine{816 } \DoxyCodeLine{817 a = u0\_l} \DoxyCodeLine{818 b = h1 * u1\_l} \DoxyCodeLine{819 c = 30.0 * um - 12.0*u0\_r - 18.0*u0\_l + 1.5*h1*(u1\_r - 3.0*u1\_l)} \DoxyCodeLine{820 d = -60.0 * um + h1 *(6.0*u1\_l - 4.0*u1\_r) + 28.0*u0\_r + 32.0*u0\_l} \DoxyCodeLine{821 e = 30.0 * um + 2.5*h1*(u1\_r - u1\_l) - 15.0*(u0\_l + u0\_r)} \DoxyCodeLine{822 } \DoxyCodeLine{823 ppoly\_coef(i1,1) = a} \DoxyCodeLine{824 ppoly\_coef(i1,2) = b} \DoxyCodeLine{825 ppoly\_coef(i1,3) = c} \DoxyCodeLine{826 ppoly\_coef(i1,4) = d} \DoxyCodeLine{827 ppoly\_coef(i1,5) = e} \DoxyCodeLine{828 } \end{DoxyCode} \mbox{\Hypertarget{namespacepqm__functions_aab9f3108956943c39ed2d4b675d78021}\label{namespacepqm__functions_aab9f3108956943c39ed2d4b675d78021}} \index{pqm\_functions@{pqm\_functions}!pqm\_limiter@{pqm\_limiter}} \index{pqm\_limiter@{pqm\_limiter}!pqm\_functions@{pqm\_functions}} \subsubsection{\texorpdfstring{pqm\_limiter()}{pqm\_limiter()}} {\footnotesize\ttfamily subroutine pqm\+\_\+functions\+::pqm\+\_\+limiter (\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)}]{edge\+\_\+slopes, }\item[{real, intent(in), optional}]{h\+\_\+neglect, }\item[{logical, intent(in), optional}]{answers\+\_\+2018 }\end{DoxyParamCaption})\hspace{0.3cm}{\ttfamily [private]}} Limit the piecewise quartic method reconstruction. Standard P\+QM limiter (White \& Adcroft, J\+CP 2008). It is assumed that the dimension of \textquotesingle{}u\textquotesingle{} is equal to the number of cells defining \textquotesingle{}grid\textquotesingle{} and \textquotesingle{}ppoly\textquotesingle{}. No consistency check is performed. \begin{DoxyParams}[1]{Parameters} \mbox{\texttt{ in}} & {\em n} & Number of cells \\ \hline \mbox{\texttt{ in}} & {\em h} & cell widths (size N) \mbox{[}H\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em u} & cell average properties (size N) \mbox{[}A\mbox{]} \\ \hline \mbox{\texttt{ in,out}} & {\em edge\+\_\+values} & Potentially modified edge values \mbox{[}A\mbox{]} \\ \hline \mbox{\texttt{ in,out}} & {\em edge\+\_\+slopes} & Potentially modified edge slopes \mbox{[}A H-\/1\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em h\+\_\+neglect} & A negligibly small width for the purpose of cell reconstructions \mbox{[}H\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em answers\+\_\+2018} & If true use older, less acccurate expressions. \\ \hline \end{DoxyParams} Definition at line 76 of file P\+Q\+M\+\_\+functions.\+F90. \begin{DoxyCode}{0} \DoxyCodeLine{76 \textcolor{keywordtype}{integer}, \textcolor{keywordtype}{intent(in)} :: N\textcolor{comment}{ !< Number of cells}} \DoxyCodeLine{77 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:)}, \textcolor{keywordtype}{intent(in)} :: h\textcolor{comment}{ !< cell widths (size N) [H]}} \DoxyCodeLine{78 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:)}, \textcolor{keywordtype}{intent(in)} :: u\textcolor{comment}{ !< cell average properties (size N) [A]}} \DoxyCodeLine{79 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:,:)}, \textcolor{keywordtype}{intent(inout)} :: edge\_values\textcolor{comment}{ !< Potentially modified edge values [A]}} \DoxyCodeLine{80 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:,:)}, \textcolor{keywordtype}{intent(inout)} :: edge\_slopes\textcolor{comment}{ !< Potentially modified edge slopes [A H-1]}} \DoxyCodeLine{81 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{optional}, \textcolor{keywordtype}{intent(in)} :: h\_neglect\textcolor{comment}{ !< A negligibly small width for}} \DoxyCodeLine{82 \textcolor{comment}{ !! the purpose of cell reconstructions [H]}} \DoxyCodeLine{83 \textcolor{keywordtype}{logical}, \textcolor{keywordtype}{optional}, \textcolor{keywordtype}{intent(in)} :: answers\_2018\textcolor{comment}{ !< If true use older, less acccurate expressions.}} \DoxyCodeLine{84 } \DoxyCodeLine{85 \textcolor{comment}{! Local variables}} \DoxyCodeLine{86 \textcolor{keywordtype}{integer} :: k \textcolor{comment}{! loop index}} \DoxyCodeLine{87 \textcolor{keywordtype}{integer} :: inflexion\_l} \DoxyCodeLine{88 \textcolor{keywordtype}{integer} :: inflexion\_r} \DoxyCodeLine{89 \textcolor{keywordtype}{ real} :: u0\_l, u0\_r \textcolor{comment}{! edge values [A]}} \DoxyCodeLine{90 \textcolor{keywordtype}{ real} :: u1\_l, u1\_r \textcolor{comment}{! edge slopes [A H-1]}} \DoxyCodeLine{91 \textcolor{keywordtype}{ real} :: u\_l, u\_c, u\_r \textcolor{comment}{! left, center and right cell averages [A]}} \DoxyCodeLine{92 \textcolor{keywordtype}{ real} :: h\_l, h\_c, h\_r \textcolor{comment}{! left, center and right cell widths [H]}} \DoxyCodeLine{93 \textcolor{keywordtype}{ real} :: sigma\_l, sigma\_c, sigma\_r \textcolor{comment}{! left, center and right van Leer slopes}} \DoxyCodeLine{94 \textcolor{keywordtype}{ real} :: slope \textcolor{comment}{! retained PLM slope}} \DoxyCodeLine{95 \textcolor{keywordtype}{ real} :: a, b, c, d, e} \DoxyCodeLine{96 \textcolor{keywordtype}{ real} :: alpha1, alpha2, alpha3} \DoxyCodeLine{97 \textcolor{keywordtype}{ real} :: rho, sqrt\_rho} \DoxyCodeLine{98 \textcolor{keywordtype}{ real} :: gradient1, gradient2} \DoxyCodeLine{99 \textcolor{keywordtype}{ real} :: x1, x2} \DoxyCodeLine{100 \textcolor{keywordtype}{ real} :: hNeglect} \DoxyCodeLine{101 } \DoxyCodeLine{102 hneglect = hneglect\_dflt ; \textcolor{keywordflow}{if} (\textcolor{keyword}{present}(h\_neglect)) hneglect = h\_neglect} \DoxyCodeLine{103 } \DoxyCodeLine{104 \textcolor{comment}{! Bound edge values}} \DoxyCodeLine{105 \textcolor{keyword}{call }bound\_edge\_values( n, h, u, edge\_values, hneglect, answers\_2018 )} \DoxyCodeLine{106 } \DoxyCodeLine{107 \textcolor{comment}{! Make discontinuous edge values monotonic (thru averaging)}} \DoxyCodeLine{108 \textcolor{keyword}{call }check\_discontinuous\_edge\_values( n, u, edge\_values )} \DoxyCodeLine{109 } \DoxyCodeLine{110 \textcolor{comment}{! Loop on interior cells to apply the PQM limiter}} \DoxyCodeLine{111 \textcolor{keywordflow}{do} k = 2,n-1} \DoxyCodeLine{112 } \DoxyCodeLine{113 \textcolor{comment}{!if ( h(k) < 1.0 ) cycle}} \DoxyCodeLine{114 } \DoxyCodeLine{115 inflexion\_l = 0} \DoxyCodeLine{116 inflexion\_r = 0} \DoxyCodeLine{117 } \DoxyCodeLine{118 \textcolor{comment}{! Get edge values, edge slopes and cell width}} \DoxyCodeLine{119 u0\_l = edge\_values(k,1)} \DoxyCodeLine{120 u0\_r = edge\_values(k,2)} \DoxyCodeLine{121 u1\_l = edge\_slopes(k,1)} \DoxyCodeLine{122 u1\_r = edge\_slopes(k,2)} \DoxyCodeLine{123 } \DoxyCodeLine{124 \textcolor{comment}{! Get cell widths and cell averages (boundary cells are assumed to}} \DoxyCodeLine{125 \textcolor{comment}{! be local extrema for the sake of slopes)}} \DoxyCodeLine{126 h\_l = h(k-1)} \DoxyCodeLine{127 h\_c = h(k)} \DoxyCodeLine{128 h\_r = h(k+1)} \DoxyCodeLine{129 u\_l = u(k-1)} \DoxyCodeLine{130 u\_c = u(k)} \DoxyCodeLine{131 u\_r = u(k+1)} \DoxyCodeLine{132 } \DoxyCodeLine{133 \textcolor{comment}{! Compute limited slope}} \DoxyCodeLine{134 sigma\_l = 2.0 * ( u\_c - u\_l ) / ( h\_c + hneglect )} \DoxyCodeLine{135 sigma\_c = 2.0 * ( u\_r - u\_l ) / ( h\_l + 2.0*h\_c + h\_r + hneglect )} \DoxyCodeLine{136 sigma\_r = 2.0 * ( u\_r - u\_c ) / ( h\_c + hneglect )} \DoxyCodeLine{137 } \DoxyCodeLine{138 \textcolor{keywordflow}{if} ( (sigma\_l * sigma\_r) > 0.0 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{139 slope = sign( min(abs(sigma\_l),abs(sigma\_c),abs(sigma\_r)), sigma\_c )} \DoxyCodeLine{140 \textcolor{keywordflow}{else}} \DoxyCodeLine{141 slope = 0.0} \DoxyCodeLine{142 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{143 } \DoxyCodeLine{144 \textcolor{comment}{! If one of the slopes has the wrong sign compared with the}} \DoxyCodeLine{145 \textcolor{comment}{! limited PLM slope, it is set equal to the limited PLM slope}} \DoxyCodeLine{146 \textcolor{keywordflow}{if} ( u1\_l*slope <= 0.0 ) u1\_l = slope} \DoxyCodeLine{147 \textcolor{keywordflow}{if} ( u1\_r*slope <= 0.0 ) u1\_r = slope} \DoxyCodeLine{148 } \DoxyCodeLine{149 \textcolor{comment}{! Local extremum --> flatten}} \DoxyCodeLine{150 \textcolor{keywordflow}{if} ( (u0\_r - u\_c) * (u\_c - u0\_l) <= 0.0) \textcolor{keywordflow}{then}} \DoxyCodeLine{151 u0\_l = u\_c} \DoxyCodeLine{152 u0\_r = u\_c} \DoxyCodeLine{153 u1\_l = 0.0} \DoxyCodeLine{154 u1\_r = 0.0} \DoxyCodeLine{155 inflexion\_l = -1} \DoxyCodeLine{156 inflexion\_r = -1} \DoxyCodeLine{157 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{158 } \DoxyCodeLine{159 \textcolor{comment}{! Edge values are bounded and averaged when discontinuous and not}} \DoxyCodeLine{160 \textcolor{comment}{! monotonic, edge slopes are consistent and the cell is not an extremum.}} \DoxyCodeLine{161 \textcolor{comment}{! We now need to check and encorce the monotonicity of the quartic within}} \DoxyCodeLine{162 \textcolor{comment}{! the cell}} \DoxyCodeLine{163 \textcolor{keywordflow}{if} ( (inflexion\_l == 0) .AND. (inflexion\_r == 0) ) \textcolor{keywordflow}{then}} \DoxyCodeLine{164 } \DoxyCodeLine{165 a = u0\_l} \DoxyCodeLine{166 b = h\_c * u1\_l} \DoxyCodeLine{167 c = 30.0 * u(k) - 12.0*u0\_r - 18.0*u0\_l + 1.5*h\_c*(u1\_r - 3.0*u1\_l)} \DoxyCodeLine{168 d = -60.0 * u(k) + h\_c *(6.0*u1\_l - 4.0*u1\_r) + 28.0*u0\_r + 32.0*u0\_l} \DoxyCodeLine{169 e = 30.0 * u(k) + 2.5*h\_c*(u1\_r - u1\_l) - 15.0*(u0\_l + u0\_r)} \DoxyCodeLine{170 } \DoxyCodeLine{171 \textcolor{comment}{! Determine the coefficients of the second derivative}} \DoxyCodeLine{172 \textcolor{comment}{! alpha1 xi\string^2 + alpha2 xi + alpha3}} \DoxyCodeLine{173 alpha1 = 6*e} \DoxyCodeLine{174 alpha2 = 3*d} \DoxyCodeLine{175 alpha3 = c} \DoxyCodeLine{176 } \DoxyCodeLine{177 rho = alpha2 * alpha2 - 4.0 * alpha1 * alpha3} \DoxyCodeLine{178 } \DoxyCodeLine{179 \textcolor{comment}{! Check whether inflexion points exist}} \DoxyCodeLine{180 \textcolor{keywordflow}{if} (( alpha1 /= 0.0 ) .and. ( rho >= 0.0 )) \textcolor{keywordflow}{then}} \DoxyCodeLine{181 } \DoxyCodeLine{182 sqrt\_rho = sqrt( rho )} \DoxyCodeLine{183 } \DoxyCodeLine{184 x1 = 0.5 * ( - alpha2 - sqrt\_rho ) / alpha1} \DoxyCodeLine{185 x2 = 0.5 * ( - alpha2 + sqrt\_rho ) / alpha1} \DoxyCodeLine{186 } \DoxyCodeLine{187 \textcolor{comment}{! Check whether both inflexion points lie in [0,1]}} \DoxyCodeLine{188 \textcolor{keywordflow}{if} ( (x1 >= 0.0) .AND. (x1 <= 1.0) .AND. \&} \DoxyCodeLine{189 (x2 >= 0.0) .AND. (x2 <= 1.0) ) \textcolor{keywordflow}{then}} \DoxyCodeLine{190 } \DoxyCodeLine{191 gradient1 = 4.0 * e * (x1**3) + 3.0 * d * (x1**2) + 2.0 * c * x1 + b} \DoxyCodeLine{192 gradient2 = 4.0 * e * (x2**3) + 3.0 * d * (x2**2) + 2.0 * c * x2 + b} \DoxyCodeLine{193 } \DoxyCodeLine{194 \textcolor{comment}{! Check whether one of the gradients is inconsistent}} \DoxyCodeLine{195 \textcolor{keywordflow}{if} ( (gradient1 * slope < 0.0) .OR. \&} \DoxyCodeLine{196 (gradient2 * slope < 0.0) ) \textcolor{keywordflow}{then}} \DoxyCodeLine{197 \textcolor{comment}{! Decide where to collapse inflexion points}} \DoxyCodeLine{198 \textcolor{comment}{! (depends on one-sided slopes)}} \DoxyCodeLine{199 \textcolor{keywordflow}{if} ( abs(sigma\_l) < abs(sigma\_r) ) \textcolor{keywordflow}{then}} \DoxyCodeLine{200 inflexion\_l = 1} \DoxyCodeLine{201 \textcolor{keywordflow}{else}} \DoxyCodeLine{202 inflexion\_r = 1} \DoxyCodeLine{203 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{204 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{205 } \DoxyCodeLine{206 \textcolor{comment}{! If both x1 and x2 do not lie in [0,1], check whether}} \DoxyCodeLine{207 \textcolor{comment}{! only x1 lies in [0,1]}} \DoxyCodeLine{208 \textcolor{keywordflow}{elseif} ( (x1 >= 0.0) .AND. (x1 <= 1.0) ) \textcolor{keywordflow}{then}} \DoxyCodeLine{209 } \DoxyCodeLine{210 gradient1 = 4.0 * e * (x1**3) + 3.0 * d * (x1**2) + 2.0 * c * x1 + b} \DoxyCodeLine{211 } \DoxyCodeLine{212 \textcolor{comment}{! Check whether the gradient is inconsistent}} \DoxyCodeLine{213 \textcolor{keywordflow}{if} ( gradient1 * slope < 0.0 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{214 \textcolor{comment}{! Decide where to collapse inflexion points}} \DoxyCodeLine{215 \textcolor{comment}{! (depends on one-sided slopes)}} \DoxyCodeLine{216 \textcolor{keywordflow}{if} ( abs(sigma\_l) < abs(sigma\_r) ) \textcolor{keywordflow}{then}} \DoxyCodeLine{217 inflexion\_l = 1} \DoxyCodeLine{218 \textcolor{keywordflow}{else}} \DoxyCodeLine{219 inflexion\_r = 1} \DoxyCodeLine{220 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{221 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{222 } \DoxyCodeLine{223 \textcolor{comment}{! If x1 does not lie in [0,1], check whether x2 lies in [0,1]}} \DoxyCodeLine{224 \textcolor{keywordflow}{elseif} ( (x2 >= 0.0) .AND. (x2 <= 1.0) ) \textcolor{keywordflow}{then}} \DoxyCodeLine{225 } \DoxyCodeLine{226 gradient2 = 4.0 * e * (x2**3) + 3.0 * d * (x2**2) + 2.0 * c * x2 + b} \DoxyCodeLine{227 } \DoxyCodeLine{228 \textcolor{comment}{! Check whether the gradient is inconsistent}} \DoxyCodeLine{229 \textcolor{keywordflow}{if} ( gradient2 * slope < 0.0 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{230 \textcolor{comment}{! Decide where to collapse inflexion points}} \DoxyCodeLine{231 \textcolor{comment}{! (depends on one-sided slopes)}} \DoxyCodeLine{232 \textcolor{keywordflow}{if} ( abs(sigma\_l) < abs(sigma\_r) ) \textcolor{keywordflow}{then}} \DoxyCodeLine{233 inflexion\_l = 1} \DoxyCodeLine{234 \textcolor{keywordflow}{else}} \DoxyCodeLine{235 inflexion\_r = 1} \DoxyCodeLine{236 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{237 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{238 } \DoxyCodeLine{239 \textcolor{keywordflow}{ endif} \textcolor{comment}{! end checking where the inflexion points lie}} \DoxyCodeLine{240 } \DoxyCodeLine{241 \textcolor{keywordflow}{ endif} \textcolor{comment}{! end checking if alpha1 != 0 AND rho >= 0}} \DoxyCodeLine{242 } \DoxyCodeLine{243 \textcolor{comment}{! If alpha1 is zero, the second derivative of the quartic reduces}} \DoxyCodeLine{244 \textcolor{comment}{! to a straight line}} \DoxyCodeLine{245 \textcolor{keywordflow}{if} (( alpha1 == 0.0 ) .and. ( alpha2 /= 0.0 )) \textcolor{keywordflow}{then}} \DoxyCodeLine{246 } \DoxyCodeLine{247 x1 = - alpha3 / alpha2} \DoxyCodeLine{248 \textcolor{keywordflow}{if} ( (x1 >= 0.0) .AND. (x1 <= 1.0) ) \textcolor{keywordflow}{then}} \DoxyCodeLine{249 } \DoxyCodeLine{250 gradient1 = 4.0 * e * (x1**3) + 3.0 * d * (x1**2) + 2.0 * c * x1 + b} \DoxyCodeLine{251 } \DoxyCodeLine{252 \textcolor{comment}{! Check whether the gradient is inconsistent}} \DoxyCodeLine{253 \textcolor{keywordflow}{if} ( gradient1 * slope < 0.0 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{254 \textcolor{comment}{! Decide where to collapse inflexion points}} \DoxyCodeLine{255 \textcolor{comment}{! (depends on one-sided slopes)}} \DoxyCodeLine{256 \textcolor{keywordflow}{if} ( abs(sigma\_l) < abs(sigma\_r) ) \textcolor{keywordflow}{then}} \DoxyCodeLine{257 inflexion\_l = 1} \DoxyCodeLine{258 \textcolor{keywordflow}{else}} \DoxyCodeLine{259 inflexion\_r = 1} \DoxyCodeLine{260 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{261 \textcolor{keywordflow}{ endif} \textcolor{comment}{! check slope consistency}} \DoxyCodeLine{262 } \DoxyCodeLine{263 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{264 } \DoxyCodeLine{265 \textcolor{keywordflow}{ endif} \textcolor{comment}{! end check whether we can find the root of the straight line}} \DoxyCodeLine{266 } \DoxyCodeLine{267 \textcolor{keywordflow}{ endif} \textcolor{comment}{! end checking whether to shift inflexion points}} \DoxyCodeLine{268 } \DoxyCodeLine{269 \textcolor{comment}{! At this point, we know onto which edge to shift inflexion points}} \DoxyCodeLine{270 \textcolor{keywordflow}{if} ( inflexion\_l == 1 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{271 } \DoxyCodeLine{272 \textcolor{comment}{! We modify the edge slopes so that both inflexion points}} \DoxyCodeLine{273 \textcolor{comment}{! collapse onto the left edge}} \DoxyCodeLine{274 u1\_l = ( 10.0 * u\_c - 2.0 * u0\_r - 8.0 * u0\_l ) / (3.0*h\_c + hneglect )} \DoxyCodeLine{275 u1\_r = ( -10.0 * u\_c + 6.0 * u0\_r + 4.0 * u0\_l ) / ( h\_c + hneglect )} \DoxyCodeLine{276 } \DoxyCodeLine{277 \textcolor{comment}{! One of the modified slopes might be inconsistent. When that happens,}} \DoxyCodeLine{278 \textcolor{comment}{! the inconsistent slope is set equal to zero and the opposite edge value}} \DoxyCodeLine{279 \textcolor{comment}{! and edge slope are modified in compliance with the fact that both}} \DoxyCodeLine{280 \textcolor{comment}{! inflexion points must still be located on the left edge}} \DoxyCodeLine{281 \textcolor{keywordflow}{if} ( u1\_l * slope < 0.0 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{282 } \DoxyCodeLine{283 u1\_l = 0.0} \DoxyCodeLine{284 u0\_r = 5.0 * u\_c - 4.0 * u0\_l} \DoxyCodeLine{285 u1\_r = 20.0 * (u\_c - u0\_l) / ( h\_c + hneglect )} \DoxyCodeLine{286 } \DoxyCodeLine{287 \textcolor{keywordflow}{elseif} ( u1\_r * slope < 0.0 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{288 } \DoxyCodeLine{289 u1\_r = 0.0} \DoxyCodeLine{290 u0\_l = (5.0*u\_c - 3.0*u0\_r) / 2.0} \DoxyCodeLine{291 u1\_l = 10.0 * (-u\_c + u0\_r) / (3.0 * h\_c + hneglect)} \DoxyCodeLine{292 } \DoxyCodeLine{293 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{294 } \DoxyCodeLine{295 \textcolor{keywordflow}{elseif} ( inflexion\_r == 1 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{296 } \DoxyCodeLine{297 \textcolor{comment}{! We modify the edge slopes so that both inflexion points}} \DoxyCodeLine{298 \textcolor{comment}{! collapse onto the right edge}} \DoxyCodeLine{299 u1\_r = ( -10.0 * u\_c + 8.0 * u0\_r + 2.0 * u0\_l ) / (3.0 * h\_c + hneglect)} \DoxyCodeLine{300 u1\_l = ( 10.0 * u\_c - 4.0 * u0\_r - 6.0 * u0\_l ) / (h\_c + hneglect)} \DoxyCodeLine{301 } \DoxyCodeLine{302 \textcolor{comment}{! One of the modified slopes might be inconsistent. When that happens,}} \DoxyCodeLine{303 \textcolor{comment}{! the inconsistent slope is set equal to zero and the opposite edge value}} \DoxyCodeLine{304 \textcolor{comment}{! and edge slope are modified in compliance with the fact that both}} \DoxyCodeLine{305 \textcolor{comment}{! inflexion points must still be located on the right edge}} \DoxyCodeLine{306 \textcolor{keywordflow}{if} ( u1\_l * slope < 0.0 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{307 } \DoxyCodeLine{308 u1\_l = 0.0} \DoxyCodeLine{309 u0\_r = ( 5.0 * u\_c - 3.0 * u0\_l ) / 2.0} \DoxyCodeLine{310 u1\_r = 10.0 * (u\_c - u0\_l) / (3.0 * h\_c + hneglect)} \DoxyCodeLine{311 } \DoxyCodeLine{312 \textcolor{keywordflow}{elseif} ( u1\_r * slope < 0.0 ) \textcolor{keywordflow}{then}} \DoxyCodeLine{313 } \DoxyCodeLine{314 u1\_r = 0.0} \DoxyCodeLine{315 u0\_l = 5.0 * u\_c - 4.0 * u0\_r} \DoxyCodeLine{316 u1\_l = 20.0 * ( -u\_c + u0\_r ) / (h\_c + hneglect)} \DoxyCodeLine{317 } \DoxyCodeLine{318 \textcolor{keywordflow}{ endif}} \DoxyCodeLine{319 } \DoxyCodeLine{320 \textcolor{keywordflow}{ endif} \textcolor{comment}{! clause to check where to collapse inflexion points}} \DoxyCodeLine{321 } \DoxyCodeLine{322 \textcolor{comment}{! Save edge values and edge slopes for reconstruction}} \DoxyCodeLine{323 edge\_values(k,1) = u0\_l} \DoxyCodeLine{324 edge\_values(k,2) = u0\_r} \DoxyCodeLine{325 edge\_slopes(k,1) = u1\_l} \DoxyCodeLine{326 edge\_slopes(k,2) = u1\_r} \DoxyCodeLine{327 } \DoxyCodeLine{328 \textcolor{keywordflow}{ enddo} \textcolor{comment}{! end loop on interior cells}} \DoxyCodeLine{329 } \DoxyCodeLine{330 \textcolor{comment}{! Constant reconstruction within boundary cells}} \DoxyCodeLine{331 edge\_values(1,:) = u(1)} \DoxyCodeLine{332 edge\_slopes(1,:) = 0.0} \DoxyCodeLine{333 } \DoxyCodeLine{334 edge\_values(n,:) = u(n)} \DoxyCodeLine{335 edge\_slopes(n,:) = 0.0} \DoxyCodeLine{336 } \end{DoxyCode} \mbox{\Hypertarget{namespacepqm__functions_afa6f7b5430011f03c428b329c5f42fae}\label{namespacepqm__functions_afa6f7b5430011f03c428b329c5f42fae}} \index{pqm\_functions@{pqm\_functions}!pqm\_reconstruction@{pqm\_reconstruction}} \index{pqm\_reconstruction@{pqm\_reconstruction}!pqm\_functions@{pqm\_functions}} \subsubsection{\texorpdfstring{pqm\_reconstruction()}{pqm\_reconstruction()}} {\footnotesize\ttfamily subroutine, public pqm\+\_\+functions\+::pqm\+\_\+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)}]{edge\+\_\+slopes, }\item[{real, dimension(\+:,\+:), intent(inout)}]{ppoly\+\_\+coef, }\item[{real, intent(in), optional}]{h\+\_\+neglect, }\item[{logical, intent(in), optional}]{answers\+\_\+2018 }\end{DoxyParamCaption})} Reconstruction by quartic polynomials within each cell. It is assumed that the dimension of \textquotesingle{}u\textquotesingle{} is equal to the number of cells defining \textquotesingle{}grid\textquotesingle{} and \textquotesingle{}ppoly\textquotesingle{}. No consistency check is performed. \begin{DoxyParams}[1]{Parameters} \mbox{\texttt{ in}} & {\em n} & Number of cells \\ \hline \mbox{\texttt{ in}} & {\em h} & cell widths (size N) \mbox{[}H\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em u} & cell averages (size N) \mbox{[}A\mbox{]} \\ \hline \mbox{\texttt{ in,out}} & {\em edge\+\_\+values} & Edge value of polynomial \mbox{[}A\mbox{]} \\ \hline \mbox{\texttt{ in,out}} & {\em edge\+\_\+slopes} & Edge slope of polynomial \mbox{[}A H-\/1\mbox{]} \\ \hline \mbox{\texttt{ in,out}} & {\em ppoly\+\_\+coef} & Coefficients of polynomial, mainly \mbox{[}A\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em h\+\_\+neglect} & A negligibly small width for the purpose of cell reconstructions \mbox{[}H\mbox{]} \\ \hline \mbox{\texttt{ in}} & {\em answers\+\_\+2018} & If true use older, less acccurate expressions. \\ \hline \end{DoxyParams} Definition at line 21 of file P\+Q\+M\+\_\+functions.\+F90. \begin{DoxyCode}{0} \DoxyCodeLine{21 \textcolor{keywordtype}{integer}, \textcolor{keywordtype}{intent(in)} :: N\textcolor{comment}{ !< Number of cells}} \DoxyCodeLine{22 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:)}, \textcolor{keywordtype}{intent(in)} :: h\textcolor{comment}{ !< cell widths (size N) [H]}} \DoxyCodeLine{23 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:)}, \textcolor{keywordtype}{intent(in)} :: u\textcolor{comment}{ !< cell averages (size N) [A]}} \DoxyCodeLine{24 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:,:)}, \textcolor{keywordtype}{intent(inout)} :: edge\_values\textcolor{comment}{ !< Edge value of polynomial [A]}} \DoxyCodeLine{25 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:,:)}, \textcolor{keywordtype}{intent(inout)} :: edge\_slopes\textcolor{comment}{ !< Edge slope of polynomial [A H-1]}} \DoxyCodeLine{26 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{dimension(:,:)}, \textcolor{keywordtype}{intent(inout)} :: ppoly\_coef\textcolor{comment}{ !< Coefficients of polynomial, mainly [A]}} \DoxyCodeLine{27 \textcolor{keywordtype}{ real}, \textcolor{keywordtype}{optional}, \textcolor{keywordtype}{intent(in)} :: h\_neglect\textcolor{comment}{ !< A negligibly small width for}} \DoxyCodeLine{28 \textcolor{comment}{ !! the purpose of cell reconstructions [H]}} \DoxyCodeLine{29 \textcolor{keywordtype}{logical}, \textcolor{keywordtype}{optional}, \textcolor{keywordtype}{intent(in)} :: answers\_2018\textcolor{comment}{ !< If true use older, less acccurate expressions.}} \DoxyCodeLine{30 } \DoxyCodeLine{31 \textcolor{comment}{! Local variables}} \DoxyCodeLine{32 \textcolor{keywordtype}{integer} :: k \textcolor{comment}{! loop index}} \DoxyCodeLine{33 \textcolor{keywordtype}{ real} :: h\_c \textcolor{comment}{! cell width}} \DoxyCodeLine{34 \textcolor{keywordtype}{ real} :: u0\_l, u0\_r \textcolor{comment}{! edge values (left and right) [A]}} \DoxyCodeLine{35 \textcolor{keywordtype}{ real} :: u1\_l, u1\_r \textcolor{comment}{! edge slopes (left and right) [A H-1]}} \DoxyCodeLine{36 \textcolor{keywordtype}{ real} :: a, b, c, d, e \textcolor{comment}{! parabola coefficients}} \DoxyCodeLine{37 } \DoxyCodeLine{38 \textcolor{comment}{! PQM limiter}} \DoxyCodeLine{39 \textcolor{keyword}{call }pqm\_limiter( n, h, u, edge\_values, edge\_slopes, h\_neglect, answers\_2018 )} \DoxyCodeLine{40 } \DoxyCodeLine{41 \textcolor{comment}{! Loop on cells to construct the cubic within each cell}} \DoxyCodeLine{42 \textcolor{keywordflow}{do} k = 1,n} \DoxyCodeLine{43 } \DoxyCodeLine{44 u0\_l = edge\_values(k,1)} \DoxyCodeLine{45 u0\_r = edge\_values(k,2)} \DoxyCodeLine{46 } \DoxyCodeLine{47 u1\_l = edge\_slopes(k,1)} \DoxyCodeLine{48 u1\_r = edge\_slopes(k,2)} \DoxyCodeLine{49 } \DoxyCodeLine{50 h\_c = h(k)} \DoxyCodeLine{51 } \DoxyCodeLine{52 a = u0\_l} \DoxyCodeLine{53 b = h\_c * u1\_l} \DoxyCodeLine{54 c = 30.0 * u(k) - 12.0*u0\_r - 18.0*u0\_l + 1.5*h\_c*(u1\_r - 3.0*u1\_l)} \DoxyCodeLine{55 d = -60.0 * u(k) + h\_c *(6.0*u1\_l - 4.0*u1\_r) + 28.0*u0\_r + 32.0*u0\_l} \DoxyCodeLine{56 e = 30.0 * u(k) + 2.5*h\_c*(u1\_r - u1\_l) - 15.0*(u0\_l + u0\_r)} \DoxyCodeLine{57 } \DoxyCodeLine{58 \textcolor{comment}{! Store coefficients}} \DoxyCodeLine{59 ppoly\_coef(k,1) = a} \DoxyCodeLine{60 ppoly\_coef(k,2) = b} \DoxyCodeLine{61 ppoly\_coef(k,3) = c} \DoxyCodeLine{62 ppoly\_coef(k,4) = d} \DoxyCodeLine{63 ppoly\_coef(k,5) = e} \DoxyCodeLine{64 } \DoxyCodeLine{65 \textcolor{keywordflow}{ enddo} \textcolor{comment}{! end loop on cells}} \DoxyCodeLine{66 } \end{DoxyCode}