Interp1PrimThirdOrderMUSCLChar.c File Reference

Characteristic-based 3rd-order MUSCL scheme with Koren's limiter. More...

#include <stdio.h>
#include <stdlib.h>
#include <basic.h>
#include <arrayfunctions.h>
#include <matmult_native.h>
#include <mathfunctions.h>
#include <interpolation.h>
#include <mpivars.h>
#include <hypar.h>

Go to the source code of this file.

Macros
#define	_MINIMUM_GHOSTS_   3

Functions
int	Interp1PrimThirdOrderMUSCLChar (double *fI, double *fC, double *u, double *x, int upw, int dir, void *s, void *m, int uflag)
	3rd order MUSCL scheme with Koren's limiter (characteristic-based) on a uniform grid More...

Detailed Description

Characteristic-based 3rd-order MUSCL scheme with Koren's limiter.

Author

Debojyoti Ghosh

Definition in file Interp1PrimThirdOrderMUSCLChar.c.

Macro Definition Documentation

#define _MINIMUM_GHOSTS_   3

Minimum number of ghost points required for this interpolation method.

Definition at line 25 of file Interp1PrimThirdOrderMUSCLChar.c.

Function Documentation

int Interp1PrimThirdOrderMUSCLChar	(	double *	fI,
		double *	fC,
		double *	u,
		double *	x,
		int	upw,
		int	dir,
		void *	s,
		void *	m,
		int	uflag
	)

3rd order MUSCL scheme with Koren's limiter (characteristic-based) on a uniform grid

Computes the interpolated values of the first primitive of a function \({\bf f}\left({\bf u}\right)\) at the interfaces from the cell-centered values of the function using the 3rd order MUSCL scheme with Koren's limiter on a uniform grid. The first primitive is defined as a function \({\bf h}\left({\bf u}\right)\) that satisfies:

\begin{equation} {\bf f}\left({\bf u}\left(x\right)\right) = \frac{1}{\Delta x} \int_{x-\Delta x/2}^{x+\Delta x/2} {\bf h}\left({\bf u}\left(\zeta\right)\right)d\zeta, \end{equation}

where \(x\) is the spatial coordinate along the dimension of the interpolation. This function computes numerical approximation \(\hat{\bf f}_{j+1/2} \approx {\bf h}_{j+1/2}\) as: using the 3rd order MUSCL scheme with Koren's limiter as follows:

\begin{equation} \hat{\alpha}^k_{j+1/2} = {\alpha}^k_{j-1} + \phi \left[\frac{1}{3}\left({\alpha}^k_j-{\alpha}^k_{j-1}\right) + \frac{1}{6}\left({\alpha}^k_{j-1}-{\alpha}^k_{j-2}\right)\right] \end{equation}

where

\begin{equation} \phi = \frac {3\left({\alpha}^k_j-{\alpha}^k_{j-1}\right)\left({\alpha}^k_{j-1}-{\alpha}^k_{j-2}\right) + \epsilon} {2\left[\left({\alpha}^k_j-{\alpha}^k_{j-1}\right)-\left({\alpha}^k_{j-1}-{\alpha}^k_{j-2}\right)\right]^2 + 3\left({\alpha}^k_j-{\alpha}^k_{j-1}\right)\left({\alpha}^k_{j-1}-{\alpha}^k_{j-2}\right) + \epsilon}, \end{equation}

\(\epsilon\) is a small constant (typically \(10^{-3}\)), and

\begin{equation} \alpha^k = {\bf l}_k \cdot {\bf f},\ k=1,\cdots,n \end{equation}

is the \(k\)-th characteristic quantity, and \({\bf l}_k\) is the \(k\)-th left eigenvector, \({\bf r}_k\) is the \(k\)-th right eigenvector, and \(n\) is HyPar::nvars. The final interpolated function is computed from the interpolated characteristic quantities as:

\begin{equation} \hat{\bf f}_{j+1/2} = \sum_{k=1}^n \alpha^k_{j+1/2} {\bf r}_k \end{equation}

Implementation Notes:

• This method assumes a uniform grid in the spatial dimension corresponding to the interpolation.
• The method described above corresponds to a left-biased interpolation. The corresponding right-biased interpolation can be obtained by reflecting the equations about interface j+1/2.

• The left and right eigenvectors are computed at an averaged quantity at j+1/2. Thus, this function requires functions to compute the average state, and the left and right eigenvectors. These are provided by the physical model through

  • HyPar::GetLeftEigenvectors()
  • HyPar::GetRightEigenvectors()
  • HyPar::AveragingFunction()

  If these functions are not provided by the physical model, then a characteristic-based interpolation cannot be used.

• The function computes the interpolant for the entire grid in one call. It loops over all the grid lines along the interpolation direction and carries out the 1D interpolation along these grid lines.
• Location of cell-centers and cell interfaces along the spatial dimension of the interpolation is shown in the following figure:
  

Function arguments:

Argument	Type	Explanation
fI	double*	Array to hold the computed interpolant at the grid interfaces. This array must have the same layout as the solution, but with no ghost points. Its size should be the same as u in all dimensions, except dir (the dimension along which to interpolate) along which it should be larger by 1 (number of interfaces is 1 more than the number of interior cell centers).
fC	double*	Array with the cell-centered values of the flux function \({\bf f}\left({\bf u}\right)\). This array must have the same layout and size as the solution, with ghost points.
u	double*	The solution array \({\bf u}\) (with ghost points). If the interpolation is characteristic based, this is needed to compute the eigendecomposition. For a multidimensional problem, the layout is as follows: u is a contiguous 1D array of size (nvars*dim[0]*dim[1]*...*dim[D-1]) corresponding to the multi-dimensional solution, with the following ordering - nvars, dim[0], dim[1], ..., dim[D-1], where nvars is the number of solution components (HyPar::nvars), dim is the local size (HyPar::dim_local), D is the number of spatial dimensions.
x	double*	The grid array (with ghost points). This is used only by non-uniform-grid interpolation methods. For multidimensional problems, the layout is as follows: x is a contiguous 1D array of size (dim[0]+dim[1]+...+dim[D-1]), with the spatial coordinates along dim[0] stored from 0,...,dim[0]-1, the spatial coordinates along dim[1] stored along dim[0],...,dim[0]+dim[1]-1, and so forth.
upw	int	Upwinding direction: if positive, a left-biased interpolant will be computed; if negative, a right-biased interpolant will be computed. If the interpolation method is central, then this has no effect.
dir	int	Spatial dimension along which to interpolate (eg: 0 for 1D; 0 or 1 for 2D; 0,1 or 2 for 3D)
s	void*	Solver object of type HyPar: the following variables are needed - HyPar::ghosts, HyPar::ndims, HyPar::nvars, HyPar::dim_local.
m	void*	MPI object of type MPIVariables: this is needed only by compact interpolation method that need to solve a global implicit system across MPI ranks.
uflag	int	A flag indicating if the function being interpolated \({\bf f}\) is the solution itself \({\bf u}\) (if 1, \({\bf f}\left({\bf u}\right) \equiv {\bf u}\)).

Reference:

• van Leer, B., Towards the Ultimate Conservative Difference Scheme. 2: Monotonicity and Conservation Combined in a Second-Order Scheme, J. of Comput. Phys., 14 (4), 1974, pp.361-370, http://dx.doi.org/10.1016/0021-9991(74)90019-9
• Koren, B., A Robust Upwind Discretization Method for Advection, Diffusion and Source Terms, Centrum voor Wiskunde en Informatica, Amsterdam, 1993

Parameters

fI	Array of interpolated function values at the interfaces
fC	Array of cell-centered values of the function \({\bf f}\left({\bf u}\right)\)
u	Array of cell-centered values of the solution \({\bf u}\)
x	Grid coordinates
upw	Upwind direction (left or right biased)
dir	Spatial dimension along which to interpolation
s	Object of type HyPar containing solver-related variables
m	Object of type MPIVariables containing MPI-related variables
uflag	Flag to indicate if \(f(u) \equiv u\), i.e, if the solution is being reconstructed

Definition at line 91 of file Interp1PrimThirdOrderMUSCLChar.c.

{
  HyPar             *solver = (HyPar*) s;
  MUSCLParameters   *muscl   = (MUSCLParameters*) solver->interp;
  int               i, k, v;
  _DECLARE_IERR_;

  int ghosts = solver->ghosts;
  int ndims  = solver->ndims;
  int nvars  = solver->nvars;
  int *dim   = solver->dim_local;

  /* define some constants */
  double one_third = 1.0/3.0;
  double one_sixth = 1.0/6.0;

  /* create index and bounds for the outer loop, i.e., to loop over all 1D lines along
     dimension "dir"                                                                    */
  int indexC[ndims], indexI[ndims], index_outer[ndims], bounds_outer[ndims], bounds_inter[ndims];
  _ArrayCopy1D_(dim,bounds_outer,ndims); bounds_outer[dir] =  1;
  _ArrayCopy1D_(dim,bounds_inter,ndims); bounds_inter[dir] += 1;
  int N_outer; _ArrayProduct1D_(bounds_outer,ndims,N_outer);

  /* allocate arrays for the averaged state, eigenvectors and characteristic interpolated f */
  double R[nvars*nvars], L[nvars*nvars], uavg[nvars], fchar[nvars];

  if (upw > 0) {
#pragma omp parallel for schedule(auto) default(shared) private(i,k,v,R,L,uavg,fchar,index_outer,indexC,indexI)
    for (i=0; i<N_outer; i++) {
      _ArrayIndexnD_(ndims,i,bounds_outer,index_outer,0);
      _ArrayCopy1D_(index_outer,indexC,ndims);
      _ArrayCopy1D_(index_outer,indexI,ndims);

      for (indexI[dir] = 0; indexI[dir] < dim[dir]+1; indexI[dir]++) {

        /* 1D indices of the stencil grid points */
        int qm1,qm2,qp1;
        indexC[dir] = indexI[dir]-2; _ArrayIndex1D_(ndims,dim,indexC,ghosts,qm2);
        indexC[dir] = indexI[dir]-1; _ArrayIndex1D_(ndims,dim,indexC,ghosts,qm1);
        indexC[dir] = indexI[dir]  ; _ArrayIndex1D_(ndims,dim,indexC,ghosts,qp1);

        int p; /* 1D index of the interface */
        _ArrayIndex1D_(ndims,bounds_inter,indexI,0,p);

        /* find averaged state at this interface */
        IERR solver->AveragingFunction(uavg,&u[nvars*qm1],&u[nvars*qp1],solver->physics);
        CHECKERR(ierr);

        /* Get the left and right eigenvectors */
        IERR solver->GetLeftEigenvectors  (uavg,L,solver->physics,dir); CHECKERR(ierr);
        IERR solver->GetRightEigenvectors (uavg,R,solver->physics,dir); CHECKERR(ierr);

        /* For each characteristic field */
        for (v = 0; v < nvars; v++) {
          /* calculate the characteristic flux components along this characteristic */
          double m2, m1, p1;
          m2 = m1 = p1 = 0;
          for (k = 0; k < nvars; k++) {
            m2 += L[v*nvars+k] * fC[qm2*nvars+k];
            m1 += L[v*nvars+k] * fC[qm1*nvars+k];
            p1 += L[v*nvars+k] * fC[qp1*nvars+k];
          }
          double fdiff = p1 - m1;
          double bdiff = m1 - m2;
          double limit =  (3*fdiff*bdiff + muscl->eps) 
                        / (2*(fdiff-bdiff)*(fdiff-bdiff) + 3*fdiff*bdiff + muscl->eps);
          fchar[v] = m1 +  limit * (one_third*fdiff + one_sixth*bdiff);
        }

        /* calculate the interface u from the characteristic u */
        IERR MatVecMult(nvars,(fI+nvars*p),R,fchar); CHECKERR(ierr);
      }
    }
  } else {
#pragma omp parallel for schedule(auto) default(shared) private(i,k,v,R,L,uavg,fchar,index_outer,indexC,indexI)
    for (i=0; i<N_outer; i++) {
      _ArrayIndexnD_(ndims,i,bounds_outer,index_outer,0);
      _ArrayCopy1D_(index_outer,indexC,ndims);
      _ArrayCopy1D_(index_outer,indexI,ndims);

      for (indexI[dir] = 0; indexI[dir] < dim[dir]+1; indexI[dir]++) {

        /* 1D indices of the stencil grid points */
        int qm1,qp1,qp2;
        indexC[dir] = indexI[dir]-1; _ArrayIndex1D_(ndims,dim,indexC,ghosts,qm1);
        indexC[dir] = indexI[dir]  ; _ArrayIndex1D_(ndims,dim,indexC,ghosts,qp1);
        indexC[dir] = indexI[dir]+1; _ArrayIndex1D_(ndims,dim,indexC,ghosts,qp2);

        int p; /* 1D index of the interface */
        _ArrayIndex1D_(ndims,bounds_inter,indexI,0,p);

        /* find averaged state at this interface */
        IERR solver->AveragingFunction(uavg,&u[nvars*qm1],&u[nvars*qp1],solver->physics);
        CHECKERR(ierr);

        /* Get the left and right eigenvectors */
        IERR solver->GetLeftEigenvectors  (uavg,L,solver->physics,dir); CHECKERR(ierr);
        IERR solver->GetRightEigenvectors (uavg,R,solver->physics,dir); CHECKERR(ierr);

        /* For each characteristic field */
        for (v = 0; v < nvars; v++) {
          /* calculate the characteristic flux components along this characteristic */
          double m1, p1, p2;
          m1 = p1 = p2 = 0;
          for (k = 0; k < nvars; k++) {
            m1 += L[v*nvars+k] * fC[qm1*nvars+k];
            p1 += L[v*nvars+k] * fC[qp1*nvars+k];
            p2 += L[v*nvars+k] * fC[qp2*nvars+k];
          }
          double fdiff = p2 - p1;
          double bdiff = p1 - m1;
          double limit =  (3*fdiff*bdiff + muscl->eps) 
                        / (2*(fdiff-bdiff)*(fdiff-bdiff) + 3*fdiff*bdiff + muscl->eps);
          fchar[v] = p1 -  limit * (one_third*fdiff + one_sixth*bdiff);
        }

        /* calculate the interface u from the characteristic u */
        IERR MatVecMult(nvars,(fI+nvars*p),R,fchar); CHECKERR(ierr);
      }
    }
  }

  return(0);
}