FirstDerivativeSecondOrderNoGhosts.c File Reference

Second order finite-difference approximation to first derivative. More...

#include <stdio.h>
#include <stdlib.h>
#include <basic.h>
#include <arrayfunctions.h>
#include <firstderivative.h>
#include <mpivars.h>

Go to the source code of this file.

Macros
#define	_MINIMUM_GHOSTS_   1

Typedefs
typedef MPIVariables	MPIContext

Functions
int	FirstDerivativeSecondOrderCentralNoGhosts (double *Df, double *f, int dir, int bias, int ndims, int *dim, int ghosts, int nvars, void *m)

Detailed Description

Second order finite-difference approximation to first derivative.

Author

Ping-Hsuan Tsai

Definition in file FirstDerivativeSecondOrderNoGhosts.c.

Macro Definition Documentation

#define _MINIMUM_GHOSTS_   1

Minimum number of ghost points required.

Definition at line 19 of file FirstDerivativeSecondOrderNoGhosts.c.

Typedef Documentation

typedef MPIVariables MPIContext

Definition at line 13 of file FirstDerivativeSecondOrderNoGhosts.c.

Function Documentation

int FirstDerivativeSecondOrderCentralNoGhosts	(	double *	Df,
		double *	f,
		int	dir,
		int	bias,
		int	ndims,
		int *	dim,
		int	ghosts,
		int	nvars,
		void *	m
	)

Computes the second-order finite-difference approximation to the first derivative (Note: not divided by the grid spacing):

\begin{equation} \left(\partial f\right)_i = \left\{ \begin{array}{ll} \frac{1}{2}\left(-3f_i+4f_{i+1}-f_{i+2}\right) & i = -g \\ \frac{1}{2}\left( f_{i+1} - f_{i-1} \right) & -g+1 \leq i \leq N+g-2 \\ \frac{1}{2}\left( f_{i-2} -4f_{i-1}+3f_i \right) & i = N+g-1 \end{array}\right. \end{equation}

where \(i\) is the grid index along the spatial dimension of the derivative, \(g\) is the number of ghost points, and \(N\) is the number of grid points (not including the ghost points) in the spatial dimension of the derivative.

Notes:

• The first derivative is computed at the grid points or the cell centers.
• Df and f are serialized 1D arrays containing the function and its computed derivatives on a multi-dimensional grid. The derivative along the specified dimension dir is computed by looping through all grid lines along dir.
• Df and f must have the same number of ghost points.
• Biased stencils are used at the physical boundaries and ghost point values of f are not used. So this function can be used when the ghost values at physical boundaries are not filled. The ghost values at internal (MPI) boundaries are still needed.

Parameters

Df	Array to hold the computed first derivative (with ghost points)
f	Array containing the grid point function values whose first derivative is to be computed (with ghost points)
dir	The spatial dimension along which the derivative is computed
bias	Forward or backward differencing for non-central finite-difference schemes (-1: backward, 1: forward)
ndims	Number of spatial/coordinate dimensions
dim	Local dimensions
ghosts	Number of ghost points
nvars	Number of vector components at each grid points
m	MPI object of type MPIContext

Definition at line 42 of file FirstDerivativeSecondOrderNoGhosts.c.

{
  MPIContext* mpi = (MPIContext*) m;
  int  i, j, v;

  if ((!Df) || (!f)) {
    fprintf(stderr, "Error in FirstDerivativeSecondOrder(): input arrays not allocated.\n");
    return(1);
  }
  if (ghosts < _MINIMUM_GHOSTS_) {
    fprintf(stderr, "Error in FirstDerivativeSecondOrderCentralNoGhosts(): insufficient number of ghosts.\n");
    return(1);
  }

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

#pragma omp parallel for schedule(auto) default(shared) private(i,j,v,index_outer,indexC)
  for (j=0; j<N_outer; j++) {
    _ArrayIndexnD_(ndims,j,bounds_outer,index_outer,0);
    _ArrayCopy1D_(index_outer,indexC,ndims);
    for (i = 0; i < dim[dir]; i++) {
      int qC, qL, qR;
      indexC[dir] = i-1; _ArrayIndex1D_(ndims,dim,indexC,ghosts,qL);
      indexC[dir] = i  ; _ArrayIndex1D_(ndims,dim,indexC,ghosts,qC );
      indexC[dir] = i+1; _ArrayIndex1D_(ndims,dim,indexC,ghosts,qR);
      for (v=0; v<nvars; v++)  Df[qC*nvars+v] = 0.5 * (f[qR*nvars+v]-f[qL*nvars+v]);
    }
  }

  if (mpi->ip[dir] == 0) {
    /* left physical boundary: overwrite the leftmost value with biased finite-difference */
#pragma omp parallel for schedule(auto) default(shared) private(i,j,v,index_outer,indexC)
    for (j=0; j<N_outer; j++) {
      _ArrayIndexnD_(ndims,j,bounds_outer,index_outer,0);
      _ArrayCopy1D_(index_outer,indexC,ndims);
      i = 0;
      int qC, qR, qR2;
      indexC[dir] = i  ; _ArrayIndex1D_(ndims,dim,indexC,ghosts,qC );
      indexC[dir] = i+1; _ArrayIndex1D_(ndims,dim,indexC,ghosts,qR);
      indexC[dir] = i+2; _ArrayIndex1D_(ndims,dim,indexC,ghosts,qR2);
      for (v=0; v<nvars; v++)  Df[qC*nvars+v] = (-0.5*f[qR2*nvars+v]+2*f[qR*nvars+v]-1.5*f[qC*nvars+v]);
    }
  }

  if (mpi->ip[dir] == (mpi->iproc[dir]-1)) {
    /* right physical boundary: overwrite the rightmost value with biased finite-difference */
#pragma omp parallel for schedule(auto) default(shared) private(i,j,v,index_outer,indexC)
    for (j=0; j<N_outer; j++) {
      _ArrayIndexnD_(ndims,j,bounds_outer,index_outer,0);
      _ArrayCopy1D_(index_outer,indexC,ndims);
      i = dim[dir] - 1;
      int qC, qL, qL2;
      indexC[dir] = i-2; _ArrayIndex1D_(ndims,dim,indexC,ghosts,qL2);
      indexC[dir] = i-1; _ArrayIndex1D_(ndims,dim,indexC,ghosts,qL);
      indexC[dir] = i  ; _ArrayIndex1D_(ndims,dim,indexC,ghosts,qC );
      for (v=0; v<nvars; v++)  Df[qC*nvars+v] = (0.5*f[qL2*nvars+v]-2*f[qL*nvars+v]+1.5*f[qC*nvars+v]);
    }
  }

  return 0;
}