SecondDerivativeFourthOrder.c File Reference

4th order discretization of the second derivative More...

#include <stdio.h>
#include <stdlib.h>
#include <basic.h>
#include <arrayfunctions.h>
#include <secondderivative.h>
#include <mpivars.h>
#include <hypar.h>

Go to the source code of this file.

Macros
#define	_MINIMUM_GHOSTS_   2

Functions
int	SecondDerivativeFourthOrderCentral (double *D2f, double *f, int dir, void *s, void *m)

Detailed Description

4th order discretization of the second derivative

Author

Debojyoti Ghosh

Definition in file SecondDerivativeFourthOrder.c.

Macro Definition Documentation

#define _MINIMUM_GHOSTS_   2

Minimum number of ghost points required.

Definition at line 18 of file SecondDerivativeFourthOrder.c.

Function Documentation

int SecondDerivativeFourthOrderCentral	(	double *	D2f,
		double *	f,
		int	dir,
		void *	s,
		void *	m
	)

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

\begin{equation} \left(\partial^2 f\right)_i = -\frac{1}{12}f_{i-2} + \frac{4}{3}f_{i-1} - \frac{15}{6}f_i + \frac{4}{3}f_{i+1} - \frac{1}{12}f_{i+2} \end{equation}

where \(i\) is the grid index along the spatial dimension of the derivative.

Notes:

• The second derivative is computed at the grid points or the cell centers.
• Though the array D2f includes ghost points, the second derivative is not computed at these locations. Thus, array elements corresponding to the ghost points contain undefined values.
• D2f and f are 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.

Parameters

D2f	Array to hold the computed second derivative (with ghost points) (same size and layout as f)
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
s	Solver object of type HyPar
m	MPI object of type MPIVariables

Definition at line 34 of file SecondDerivativeFourthOrder.c.

{
  HyPar         *solver = (HyPar*) s;
  int           i, v;

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

  static double one_twelve = 1.0/12.0;

  if ((!D2f) || (!f)) {
    fprintf(stderr, "Error in SecondDerivativeFourthOrder(): input arrays not allocated.\n");
    return(1);
  }
  if (ghosts < _MINIMUM_GHOSTS_) {
    fprintf(stderr, "Error in SecondDerivativeFourthOrder(): 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 done = 0; _ArraySetValue_(index_outer,ndims,0);
  while (!done) {
    _ArrayCopy1D_(index_outer,indexC,ndims);
    for (i = 0; i < dim[dir]; i++) {
      int qm2, qm1, qC, qp1, qp2;
      indexC[dir] = i-2; _ArrayIndex1D_(ndims,dim,indexC,ghosts,qm2);
      indexC[dir] = i-1; _ArrayIndex1D_(ndims,dim,indexC,ghosts,qm1);
      indexC[dir] = i  ; _ArrayIndex1D_(ndims,dim,indexC,ghosts,qC );
      indexC[dir] = i+1; _ArrayIndex1D_(ndims,dim,indexC,ghosts,qp1);
      indexC[dir] = i+2; _ArrayIndex1D_(ndims,dim,indexC,ghosts,qp2);
      for (v=0; v<nvars; v++)  
        D2f[qC*nvars+v] = (-f[qm2*nvars+v]+16*f[qm1*nvars+v]-30*f[qC*nvars+v]+16*f[qp1*nvars+v]-f[qp2*nvars+v])*one_twelve;
    }
    _ArrayIncrementIndex_(ndims,bounds_outer,index_outer,done);
  }
  
  return(0);
}