p3m_functions real, parameter real, parameter p3m_functions::hneglect_dflt hneglect_dflt = 1.E-30 Default value of a negligible cell thickness. p3m_boundary_extrapolation p3m_limiter real, parameter real, parameter p3m_functions::hneglect_edge_dflt hneglect_edge_dflt = 1.E-10 Default value of a negligible edge thickness. p3m_boundary_extrapolation subroutine, public subroutine, public p3m_functions::p3m_interpolation (N, h, u, edge_values, ppoly_S, ppoly_coef, h_neglect, answers_2018) p3m_interpolation N N h h u u edge_values edge_values ppoly_S ppoly_S ppoly_coef ppoly_coef h_neglect h_neglect answers_2018 answers_2018 Set up a piecewise cubic interpolation from cell averages and estimated edge slopes and values. Cubic interpolation between edges. The edge values and slopes MUST have been estimated prior to calling this routine. It is assumed that the size of the array 'u' is equal to the number of cells defining 'grid' and 'ppoly'. No consistency check is performed here. n Number of cells h cell widths (size N) [H] u cell averages (size N) in arbitrary units [A] edge_values Edge value of polynomial [A] ppoly_s Edge slope of polynomial [A H-1]. ppoly_coef Coefficients of polynomial [A] h_neglect A negligibly small width for the purpose of cell reconstructions [H] answers_2018 If true use older, less acccurate expressions. p3m_limiter regrid_interp::regridding_set_ppolys subroutine subroutine p3m_functions::p3m_limiter (N, h, u, edge_values, ppoly_S, ppoly_coef, h_neglect, answers_2018) p3m_limiter N N h h u u edge_values edge_values ppoly_S ppoly_S ppoly_coef ppoly_coef h_neglect h_neglect answers_2018 answers_2018 Adust a piecewise cubic reconstruction with a limiter that adjusts the edge values and slopes. The p3m limiter operates as follows: Edge values are bounded Discontinuous edge values are systematically averaged Loop on cells and do the following a. Build cubic curve b. Check if cubic curve is monotonic c. If not, monotonize cubic curve and rebuild it Step 3 of the monotonization process leaves all edge values unchanged. n Number of cells h cell widths (size N) [H] u cell averages (size N) in arbitrary units [A] edge_values Edge value of polynomial [A] ppoly_s Edge slope of polynomial [A H-1] ppoly_coef Coefficients of polynomial [A] h_neglect A negligibly small width for the purpose of cell reconstructions [H] answers_2018 If true use older, less acccurate expressions. regrid_edge_values::average_discontinuous_edge_values regrid_edge_values::bound_edge_values build_cubic_interpolant hneglect_dflt is_cubic_monotonic monotonize_cubic p3m_interpolation subroutine, public subroutine, public p3m_functions::p3m_boundary_extrapolation (N, h, u, edge_values, ppoly_S, ppoly_coef, h_neglect, h_neglect_edge) p3m_boundary_extrapolation N N h h u u edge_values edge_values ppoly_S ppoly_S ppoly_coef ppoly_coef h_neglect h_neglect h_neglect_edge h_neglect_edge Calculate the edge values and slopes at boundary cells as part of building a piecewise cubic sub-grid scale profiles. The following explanations apply to the left boundary cell. The same reasoning holds for the right boundary cell. A cubic needs to be built in the cell and requires four degrees of freedom, which are the edge values and slopes. 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 left edge value and slope are determined by computing the parabola based on the cell average and the right edge value and slope. The resulting cubic is not necessarily monotonic and the slopes are subsequently modified to yield a monotonic cubic. n Number of cells h cell widths (size N) [H] u cell averages (size N) in arbitrary units [A] edge_values Edge value of polynomial [A] ppoly_s Edge slope of polynomial [A H-1] ppoly_coef Coefficients of polynomial [A] h_neglect A negligibly small width for the purpose of cell reconstructions [H] h_neglect_edge A negligibly small width for the purpose of finding edge values [H] build_cubic_interpolant hneglect_dflt hneglect_edge_dflt is_cubic_monotonic monotonize_cubic regrid_interp::regridding_set_ppolys subroutine subroutine p3m_functions::build_cubic_interpolant (h, k, edge_values, ppoly_S, ppoly_coef) build_cubic_interpolant h h k k edge_values edge_values ppoly_S ppoly_S ppoly_coef ppoly_coef Build cubic interpolant in cell k. Given edge values and edge slopes, compute coefficients of cubic in cell k. NOTE: edge values and slopes MUST have been properly calculated prior to calling this routine. h cell widths (size N) [H] k The index of the cell to work on edge_values Edge value of polynomial in arbitrary units [A] ppoly_s Edge slope of polynomial [A H-1] ppoly_coef Coefficients of polynomial [A] p3m_boundary_extrapolation p3m_limiter logical function logical function p3m_functions::is_cubic_monotonic (ppoly_coef, k) is_cubic_monotonic ppoly_coef ppoly_coef k k Check whether the cubic reconstruction in cell k is monotonic. This function checks whether the cubic curve in cell k is monotonic. If so, returns 1. Otherwise, returns 0. The cubic is monotonic if the first derivative is single-signed in (0,1). Hence, we check whether the roots (if any) lie inside this interval. If there is no root or if both roots lie outside this interval, the cubic is monotonic. ppoly_coef Coefficients of cubic polynomial in arbitary units [A] k The index of the cell to work on p3m_boundary_extrapolation p3m_limiter subroutine subroutine p3m_functions::monotonize_cubic (h, u0_l, u0_r, sigma_l, sigma_r, slope, u1_l, u1_r) monotonize_cubic h h u0_l u0_l u0_r u0_r sigma_l sigma_l sigma_r sigma_r slope slope u1_l u1_l u1_r u1_r Monotonize a cubic curve by modifying the edge slopes. This routine takes care of monotonizing a cubic on [0,1] by modifying the edge slopes. The edge values are NOT modified. The cubic is entirely determined by the four degrees of freedom u0_l, u0_r, u1_l and u1_r. u1_l and u1_r are the edge slopes expressed in the GLOBAL coordinate system. The monotonization occurs as follows. h cell width [H] u0_l left edge value in arbitrary units [A] u0_r right edge value [A] sigma_l left 2nd-order slopes [A H-1] sigma_r right 2nd-order slopes [A H-1] slope limited PLM slope [A H-1] u1_l left edge slopes [A H-1] u1_r right edge slopes [A H-1] p3m_boundary_extrapolation p3m_limiter Cubic interpolation functions. Date of creation: 2008.06.09 L. White This module contains p3m interpolation routines. p3m interpolation is performed by estimating the edge values and slopes and constructing a cubic polynomial. We then make sure that the edge values are bounded and continuous and we then modify the slopes to get a monotonic cubic curve.