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jon-serial.c
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/*
** Code to implement a d2q9-bgk lattice boltzmann scheme.
** 'd2' inidates a 2-dimensional grid, and
** 'q9' indicates 9 velocities per grid cell.
** 'bgk' refers to the Bhatnagar-Gross-Krook collision step.
**
** The 'speeds' in each cell are numbered as follows:
**
** 6 2 5
** \|/
** 3-0-1
** /|\
** 7 4 8
**
** A 2D grid 'unwrapped' in row major order to give a 1D array:
**
** cols
** --- --- ---
** | D | E | F |
** rows --- --- ---
** | A | B | C |
** --- --- ---
**
** --- --- --- --- --- ---
** | A | B | C | D | E | F |
** --- --- --- --- --- ---
*/
#include<stdio.h>
#include<stdlib.h>
#include<time.h>
#include<sys/time.h>
#include<omp.h>
#define NSPEEDS 9
#define PARAMFILE "input.params"
#define OBSTACLEFILE "obstacles.dat"
#define FINALSTATEFILE "final_state.dat"
#define AVVELSFILE "av_vels.dat"
/* struct to hold the parameter values */
typedef struct {
int nx; /* no. of cells in y-deirection */
int ny; /* no. of cells in x-direction */
int maxIters; /* no. of iterations */
int reynolds_dim; /* dimension for Reynolds number */
double density; /* density per link */
double accel; /* density redistribution */
double omega; /* relaxation parameter */
double accelerate_w1;
double accelerate_w2;
} t_param;
/* struct to hold the 'speed' values */
typedef struct {
float speeds[NSPEEDS];
} t_speed;
enum boolean { FALSE, TRUE };
/*
** function prototypes
*/
/*
* Extra Shiz added by Jon
*/
void timeval_subtract (struct timeval *result, struct timeval *x, struct timeval *y);
/* load params, allocate memory, load obstacles & initialise fluid particle densities */
void initialise(t_param* params, t_speed** cells_ptr, t_speed** tmp_cells_ptr,
int** obstacles_ptr, float** av_vels_ptr);
/*
** The main calculation methods.
** timestep calls, in order, the functions:
** accelerate_flow(), propagate(), rebound() & collision()
*/
void timestep(const t_param params, t_speed* cells, t_speed* tmp_cells, int* obstacles);
void accelerate_flow(const t_param params, t_speed* cells, int* obstacles);
void propagate(const t_param params, t_speed* cells, t_speed* tmp_cells);
void rebound(const t_param params, t_speed* cells, t_speed* tmp_cells, int* obstacles);
void collision(const t_param params, t_speed* cells, t_speed* tmp_cells, int* obstacles);
void write_values(const t_param params, t_speed* cells, int* obstacles, float* av_vels);
/* finalise, including freeing up allocated memory */
void finalise(const t_param* params, t_speed** cells_ptr, t_speed** tmp_cells_ptr,
int** obstacles_ptr, float** av_vels_ptr);
/* Sum all the densities in the grid.
** The total should remain constant from one timestep to the next. */
double total_density(const t_param params, t_speed* cells);
/* compute average velocity */
double av_velocity(const t_param params, t_speed* cells, int* obstacles);
/* calculate Reynolds number */
double calc_reynolds(const t_param params, t_speed* cells, int* obstacles);
/* utility functions */
void die(const char* message, const int line, const char *file);
/*
** main program:
** initialise, timestep loop, finalise
*/
int main(int argc, char* argv[])
{
t_param params; /* struct to hold parameter values */
t_speed* cells = NULL; /* grid containing fluid densities */
t_speed* tmp_cells = NULL; /* scratch space */
int* obstacles = NULL; /* grid indicating which cells are blocked */
float* av_vels = NULL; /* a record of the av. velocity computed for each timestep */
int ii; /* generic counter */
struct timeval start_time;
struct timeval end_time;
struct timeval result_time;
/* initialise our data structures and load values from file */
initialise(¶ms, &cells, &tmp_cells, &obstacles, &av_vels);
/* iterate for maxIters timesteps */
gettimeofday(&start_time, NULL);
for (ii=0;ii<params.maxIters;ii++) {
timestep(params,cells,tmp_cells,obstacles);
av_vels[ii] = av_velocity(params,cells,obstacles);
#ifdef DEBUG
printf("==timestep: %d==\n",ii);
printf("av velocity: %.12E\n", av_vels[ii]);
printf("tot density: %.12E\n",total_density(params,cells));
#endif
}
gettimeofday(&end_time, NULL);
timeval_subtract(&result_time, &end_time, &start_time);
/* write final values and free memory */
printf("==done==\n");
printf("Reynolds number:\t%.12E\n",calc_reynolds(params,cells,obstacles));
printf("Elapsed gettimeofday:\t\t%d s %d us\n", result_time.tv_sec, result_time.tv_usec);
write_values(params,cells,obstacles,av_vels);
finalise(¶ms, &cells, &tmp_cells, &obstacles, &av_vels);
return EXIT_SUCCESS;
}
void timeval_subtract (struct timeval *result, struct timeval *x, struct timeval *y)
{
/* Perform the carry for the later subtraction by updating y. */
if (x->tv_usec < y->tv_usec) {
int nsec = (y->tv_usec - x->tv_usec) / 1000000 + 1;
y->tv_usec -= 1000000 * nsec;
y->tv_sec += nsec;
}
if (x->tv_usec - y->tv_usec > 1000000) {
int nsec = (x->tv_usec - y->tv_usec) / 1000000;
y->tv_usec += 1000000 * nsec;
y->tv_sec -= nsec;
}
result->tv_sec = x->tv_sec - y->tv_sec;
result->tv_usec = x->tv_usec - y->tv_usec;
}
void timestep(const t_param params, t_speed* cells, t_speed* tmp_cells, int* obstacles)
{
accelerate_flow(params,cells,obstacles);
propagate(params,cells,tmp_cells);
rebound(params,cells,tmp_cells,obstacles);
collision(params,cells,tmp_cells,obstacles);
}
void accelerate_flow(const t_param params, t_speed* cells, int* obstacles)
{
unsigned int ii,jj; /* generic counters */
/* modify the first column of the grid */
jj=0;
//#pragma omp parallel for schedule(static) private(ii) firstprivate(jj) shared(cells)
for(ii=0;ii<params.ny;ii++) {
/* if the cell is not occupied and
** we don't send a density negative */
if( !obstacles[ii*params.nx + jj] )
{
t_speed temp_cell = cells[ii*params.nx + jj];
if ((temp_cell.speeds[3] - params.accelerate_w1) > 0.0 &&
(temp_cell.speeds[6] - params.accelerate_w2) > 0.0 &&
(temp_cell.speeds[7] - params.accelerate_w2) > 0.0 )
{
/* increase 'east-side' densities */
temp_cell.speeds[1] += params.accelerate_w1;
temp_cell.speeds[5] += params.accelerate_w2;
temp_cell.speeds[8] += params.accelerate_w2;
/* decrease 'west-side' densities */
temp_cell.speeds[3] -= params.accelerate_w1;
temp_cell.speeds[6] -= params.accelerate_w2;
temp_cell.speeds[7] -= params.accelerate_w2;
}
cells[ii*params.nx + jj] = temp_cell;
}
}
}
void propagate(const t_param params, t_speed* cells, t_speed* tmp_cells)
{
unsigned int ii,jj; /* generic counters */
unsigned int x_e,x_w,y_n,y_s; /* indices of neighbouring cells */
/* loop over _all_ cells */
//#pragma omp parallel for schedule(static) private(ii, jj, x_e, x_w, y_n, y_s) shared(cells, tmp_cells)
for(ii=0;ii<params.ny;ii++) {
for(jj=0;jj<params.nx;jj++) {
/* determine indices of axis-direction neighbours
** respecting periodic boundary conditions (wrap around) */
y_n = (ii + 1) % params.ny;
x_e = (jj + 1) % params.nx;
y_s = (ii == 0) ? (ii + params.ny - 1) : (ii - 1);
x_w = (jj == 0) ? (jj + params.nx - 1) : (jj - 1);
/* propagate densities to neighbouring cells, following
** appropriate directions of travel and writing into
** scratch space grid */
t_speed temp_cell = cells[ii*params.nx + jj];
tmp_cells[ii *params.nx + jj].speeds[0] = temp_cell.speeds[0]; /* central cell, */
/* no movement */
tmp_cells[ii *params.nx + x_e].speeds[1] = temp_cell.speeds[1]; /* east */
tmp_cells[y_n*params.nx + jj].speeds[2] = temp_cell.speeds[2]; /* north */
tmp_cells[ii *params.nx + x_w].speeds[3] = temp_cell.speeds[3]; /* west */
tmp_cells[y_s*params.nx + jj].speeds[4] = temp_cell.speeds[4]; /* south */
tmp_cells[y_n*params.nx + x_e].speeds[5] = temp_cell.speeds[5]; /* north-east */
tmp_cells[y_n*params.nx + x_w].speeds[6] = temp_cell.speeds[6]; /* north-west */
tmp_cells[y_s*params.nx + x_w].speeds[7] = temp_cell.speeds[7]; /* south-west */
tmp_cells[y_s*params.nx + x_e].speeds[8] = temp_cell.speeds[8]; /* south-east */
}
}
}
void rebound(const t_param params, t_speed* cells, t_speed* tmp_cells, int* obstacles)
{
unsigned int ii,jj; /* generic counters */
/* loop over the cells in the grid */
//#pragma omp parallel for schedule(static) private(ii, jj) shared(cells, tmp_cells)
for(ii=0;ii<params.ny;ii++) {
for(jj=0;jj<params.nx;jj++) {
/* if the cell contains an obstacle */
if(obstacles[ii*params.nx + jj]) {
/* called after propagate, so taking values from scratch space
** mirroring, and writing into main grid */
t_speed temp_tmp_cell = tmp_cells[ii*params.nx + jj];
t_speed temp_cell = cells[ii*params.nx + jj];
temp_cell.speeds[1] = temp_tmp_cell.speeds[3];
temp_cell.speeds[2] = temp_tmp_cell.speeds[4];
temp_cell.speeds[3] = temp_tmp_cell.speeds[1];
temp_cell.speeds[4] = temp_tmp_cell.speeds[2];
temp_cell.speeds[5] = temp_tmp_cell.speeds[7];
temp_cell.speeds[6] = temp_tmp_cell.speeds[8];
temp_cell.speeds[7] = temp_tmp_cell.speeds[5];
temp_cell.speeds[8] = temp_tmp_cell.speeds[6];
cells[ii*params.nx + jj] = temp_cell;
}
}
}
}
void collision(const t_param params, t_speed* cells, t_speed* tmp_cells, int* obstacles)
{
unsigned int ii,jj,kk; /* generic counters */
const float c_sq = 1.0/3.0; /* square of speed of sound */
const float w0 = 4.0/9.0; /* weighting factor */
const float w1 = 1.0/9.0; /* weighting factor */
const float w2 = 1.0/36.0; /* weighting factor */
float u_x,u_y; /* av. velocities in x and y directions */
float u[NSPEEDS]; /* directional velocities */
float d_equ[NSPEEDS]; /* equilibrium densities */
float u_sq; /* squared velocity */
float local_density; /* sum of densities in a particular cell */
/* loop over the cells in the grid
** NB the collision step is called after
** the propagate step and so values of interest
** are in the scratch-space grid */
//#pragma omp parallel for schedule(static) private(ii, jj, kk, u, u_x, u_y, d_equ, u_sq, local_density) shared(cells, tmp_cells)
for(ii=0 ;ii<params.ny;ii++)
{
for(jj=0;jj<params.nx;jj++)
{
/* don't consider occupied cells */
if(!obstacles[ii*params.nx + jj])
{
t_speed temp_cell = tmp_cells[ii*params.nx + jj];
/* compute local density total */
local_density = 0.0;
for(kk=0;kk<NSPEEDS;kk++)
{
local_density += temp_cell.speeds[kk];
}
/* compute x velocity component */
u_x = (temp_cell.speeds[1] +
temp_cell.speeds[5] +
temp_cell.speeds[8] -
(temp_cell.speeds[3] +
temp_cell.speeds[6] +
temp_cell.speeds[7]))
/ local_density;
/* compute y velocity component */
u_y = (temp_cell.speeds[2] +
temp_cell.speeds[5] +
temp_cell.speeds[6]
- (temp_cell.speeds[4] +
temp_cell.speeds[7] +
temp_cell.speeds[8]))
/ local_density;
/* velocity squared */
u_sq = u_x * u_x + u_y * u_y;
/* directional velocity components */
u[1] = u_x; /* east */
u[2] = u_y; /* north */
u[3] = - u_x; /* west */
u[4] = - u_y; /* south */
u[5] = u_x + u_y; /* north-east */
u[6] = - u_x + u_y; /* north-west */
u[7] = - u_x - u_y; /* south-west */
u[8] = u_x - u_y; /* south-east */
/* equilibrium densities */
/* zero velocity density: weight w0 */
d_equ[0] = w0 * local_density * (1.0 - u_sq / (2.0 * c_sq));
/* axis speeds: weight w1 */
d_equ[1] = w1 * local_density * (1.0 + u[1] / c_sq
+ (u[1] * u[1]) / (2.0 * c_sq * c_sq)
- u_sq / (2.0 * c_sq));
d_equ[2] = w1 * local_density * (1.0 + u[2] / c_sq
+ (u[2] * u[2]) / (2.0 * c_sq * c_sq)
- u_sq / (2.0 * c_sq));
d_equ[3] = w1 * local_density * (1.0 + u[3] / c_sq
+ (u[3] * u[3]) / (2.0 * c_sq * c_sq)
- u_sq / (2.0 * c_sq));
d_equ[4] = w1 * local_density * (1.0 + u[4] / c_sq
+ (u[4] * u[4]) / (2.0 * c_sq * c_sq)
- u_sq / (2.0 * c_sq));
/* diagonal speeds: weight w2 */
d_equ[5] = w2 * local_density * (1.0 + u[5] / c_sq
+ (u[5] * u[5]) / (2.0 * c_sq * c_sq)
- u_sq / (2.0 * c_sq));
d_equ[6] = w2 * local_density * (1.0 + u[6] / c_sq
+ (u[6] * u[6]) / (2.0 * c_sq * c_sq)
- u_sq / (2.0 * c_sq));
d_equ[7] = w2 * local_density * (1.0 + u[7] / c_sq
+ (u[7] * u[7]) / (2.0 * c_sq * c_sq)
- u_sq / (2.0 * c_sq));
d_equ[8] = w2 * local_density * (1.0 + u[8] / c_sq
+ (u[8] * u[8]) / (2.0 * c_sq * c_sq)
- u_sq / (2.0 * c_sq));
/* relaxation step */
t_speed temp_cell_for_cells = cells[ii*params.nx + jj];
for(kk=0;kk<NSPEEDS;kk++)
{
temp_cell_for_cells.speeds[kk] = (temp_cell.speeds[kk]
+ params.omega *
(d_equ[kk] - temp_cell.speeds[kk]));
}
cells[ii*params.nx + jj] = temp_cell_for_cells;
tmp_cells[ii*params.nx + jj] = temp_cell;
}
}
}
}
void initialise(t_param* params, t_speed** cells_ptr, t_speed** tmp_cells_ptr,
int** obstacles_ptr, float** av_vels_ptr)
{
FILE *fp; /* file pointer */
int ii,jj; /* generic counters */
int xx,yy; /* generic array indices */
int blocked; /* indicates whether a cell is blocked by an obstacle */
int retval; /* to hold return value for checking */
double w0,w1,w2; /* weighting factors */
/* open the parameter file */
fp = fopen(PARAMFILE,"r");
if (fp == NULL) {
die("could not open file input.params",__LINE__,__FILE__);
}
/* read in the parameter values */
retval = fscanf(fp,"%d\n",&(params->nx));
if(retval != 1) die ("could not read param file: nx",__LINE__,__FILE__);
retval = fscanf(fp,"%d\n",&(params->ny));
if(retval != 1) die ("could not read param file: ny",__LINE__,__FILE__);
retval = fscanf(fp,"%d\n",&(params->maxIters));
if(retval != 1) die ("could not read param file: maxIters",__LINE__,__FILE__);
retval = fscanf(fp,"%d\n",&(params->reynolds_dim));
if(retval != 1) die ("could not read param file: reynolds_dim",__LINE__,__FILE__);
retval = fscanf(fp,"%lf\n",&(params->density));
if(retval != 1) die ("could not read param file: density",__LINE__,__FILE__);
retval = fscanf(fp,"%lf\n",&(params->accel));
if(retval != 1) die ("could not read param file: accel",__LINE__,__FILE__);
retval = fscanf(fp,"%lf\n",&(params->omega));
if(retval != 1) die ("could not read param file: omega",__LINE__,__FILE__);
/* and close up the file */
fclose(fp);
// Calculate these here for accelerate function only once:
params->accelerate_w1 = params->density * params->accel / 9.0;
params->accelerate_w2 = params->density * params->accel / 36.0;
/*
** Allocate memory.
**
** Remember C is pass-by-value, so we need to
** pass pointers into the initialise function.
**
** NB we are allocating a 1D array, so that the
** memory will be contiguous. We still want to
** index this memory as if it were a (row major
** ordered) 2D array, however. We will perform
** some arithmetic using the row and column
** coordinates, inside the square brackets, when
** we want to access elements of this array.
**
** Note also that we are using a structure to
** hold an array of 'speeds'. We will allocate
** a 1D array of these structs.
*/
/* main grid */
*cells_ptr = (t_speed*)malloc(sizeof(t_speed)*(params->ny*params->nx));
if (*cells_ptr == NULL)
die("cannot allocate memory for cells",__LINE__,__FILE__);
/* 'helper' grid, used as scratch space */
*tmp_cells_ptr = (t_speed*)malloc(sizeof(t_speed)*(params->ny*params->nx));
if (*tmp_cells_ptr == NULL)
die("cannot allocate memory for tmp_cells",__LINE__,__FILE__);
/* the map of obstacles */
*obstacles_ptr = malloc(sizeof(int*)*(params->ny*params->nx));
if (*obstacles_ptr == NULL)
die("cannot allocate column memory for obstacles",__LINE__,__FILE__);
/* initialise densities */
w0 = params->density * 4.0/9.0;
w1 = params->density /9.0;
w2 = params->density /36.0;
for(ii=0;ii<params->ny;ii++) {
for(jj=0;jj<params->nx;jj++) {
/* centre */
(*cells_ptr)[ii*params->nx + jj].speeds[0] = w0;
/* axis directions */
(*cells_ptr)[ii*params->nx + jj].speeds[1] = w1;
(*cells_ptr)[ii*params->nx + jj].speeds[2] = w1;
(*cells_ptr)[ii*params->nx + jj].speeds[3] = w1;
(*cells_ptr)[ii*params->nx + jj].speeds[4] = w1;
/* diagonals */
(*cells_ptr)[ii*params->nx + jj].speeds[5] = w2;
(*cells_ptr)[ii*params->nx + jj].speeds[6] = w2;
(*cells_ptr)[ii*params->nx + jj].speeds[7] = w2;
(*cells_ptr)[ii*params->nx + jj].speeds[8] = w2;
}
}
/* first set all cells in obstacle array to zero */
for(ii=0;ii<params->ny;ii++) {
for(jj=0;jj<params->nx;jj++) {
(*obstacles_ptr)[ii*params->nx + jj] = 0;
}
}
/* open the obstacle data file */
fp = fopen(OBSTACLEFILE,"r");
if (fp == NULL) {
die("could not open file obstacles",__LINE__,__FILE__);
}
/* read-in the blocked cells list */
while( (retval = fscanf(fp,"%d %d %d\n", &xx, &yy, &blocked)) != EOF) {
/* some checks */
if ( retval != 3)
die("expected 3 values per line in obstacle file",__LINE__,__FILE__);
if ( xx<0 || xx>params->nx-1 )
die("obstacle x-coord out of range",__LINE__,__FILE__);
if ( yy<0 || yy>params->ny-1 )
die("obstacle y-coord out of range",__LINE__,__FILE__);
if ( blocked != 1 )
die("obstacle blocked value should be 1",__LINE__,__FILE__);
/* assign to array */
(*obstacles_ptr)[yy*params->nx + xx] = blocked;
}
/* and close the file */
fclose(fp);
/*
** allocate space to hold a record of the avarage velocities computed
** at each timestep
*/
*av_vels_ptr = (double*)malloc(sizeof(double)*params->maxIters);
}
void finalise(const t_param* params, t_speed** cells_ptr, t_speed** tmp_cells_ptr,
int** obstacles_ptr, float** av_vels_ptr)
{
/*
** free up allocated memory
*/
free(*cells_ptr);
*cells_ptr = NULL;
free(*tmp_cells_ptr);
*tmp_cells_ptr = NULL;
free(*obstacles_ptr);
*obstacles_ptr = NULL;
free(*av_vels_ptr);
*av_vels_ptr = NULL;
}
double av_velocity(const t_param params, t_speed* cells, int* obstacles)
{
int ii,jj,kk; /* generic counters */
int tot_cells = 0; /* no. of cells used in calculation */
double local_density; /* total density in cell */
double tot_u_x; /* accumulated x-components of velocity */
/* initialise */
tot_u_x = 0.0;
/* loop over all non-blocked cells */
for(ii=0;ii<params.ny;ii++) {
for(jj=0;jj<params.nx;jj++) {
/* ignore occupied cells */
if(!obstacles[ii*params.nx + jj]) {
/* local density total */
local_density = 0.0;
for(kk=0;kk<NSPEEDS;kk++) {
local_density += cells[ii*params.nx + jj].speeds[kk];
}
/* x-component of velocity */
tot_u_x += (cells[ii*params.nx + jj].speeds[1] +
cells[ii*params.nx + jj].speeds[5] +
cells[ii*params.nx + jj].speeds[8]
- (cells[ii*params.nx + jj].speeds[3] +
cells[ii*params.nx + jj].speeds[6] +
cells[ii*params.nx + jj].speeds[7])) /
local_density;
/* increase counter of inspected cells */
++tot_cells;
}
}
}
return tot_u_x / (double)tot_cells;
}
double calc_reynolds(const t_param params, t_speed* cells, int* obstacles)
{
const double viscosity = 1.0 / 6.0 * (2.0 / params.omega - 1.0);
return av_velocity(params,cells,obstacles) * params.reynolds_dim / viscosity;
}
double total_density(const t_param params, t_speed* cells)
{
int ii,jj,kk; /* generic counters */
double total = 0.0; /* accumulator */
for(ii=0;ii<params.ny;ii++) {
for(jj=0;jj<params.nx;jj++) {
for(kk=0;kk<NSPEEDS;kk++) {
total += cells[ii*params.nx + jj].speeds[kk];
}
}
}
return total;
}
void write_values(const t_param params, t_speed* cells, int* obstacles, float* av_vels)
{
FILE* fp; /* file pointer */
int ii,jj,kk; /* generic counters */
const double c_sq = 1.0/3.0; /* sq. of speed of sound */
double local_density; /* per grid cell sum of densities */
double pressure; /* fluid pressure in grid cell */
double u_x; /* x-component of velocity in grid cell */
double u_y; /* y-component of velocity in grid cell */
fp = fopen(FINALSTATEFILE,"w");
if (fp == NULL) {
die("could not open file output file",__LINE__,__FILE__);
}
for(ii=0;ii<params.ny;ii++) {
for(jj=0;jj<params.nx;jj++) {
/* an occupied cell */
if(obstacles[ii*params.nx + jj]) {
u_x = u_y = 0.0;
pressure = params.density * c_sq;
}
/* no obstacle */
else {
local_density = 0.0;
for(kk=0;kk<NSPEEDS;kk++) {
local_density += cells[ii*params.nx + jj].speeds[kk];
}
/* compute x velocity component */
u_x = (cells[ii*params.nx + jj].speeds[1] +
cells[ii*params.nx + jj].speeds[5] +
cells[ii*params.nx + jj].speeds[8]
- (cells[ii*params.nx + jj].speeds[3] +
cells[ii*params.nx + jj].speeds[6] +
cells[ii*params.nx + jj].speeds[7]))
/ local_density;
/* compute y velocity component */
u_y = (cells[ii*params.nx + jj].speeds[2] +
cells[ii*params.nx + jj].speeds[5] +
cells[ii*params.nx + jj].speeds[6]
- (cells[ii*params.nx + jj].speeds[4] +
cells[ii*params.nx + jj].speeds[7] +
cells[ii*params.nx + jj].speeds[8]))
/ local_density;
/* compute pressure */
pressure = local_density * c_sq;
}
/* write to file */
fprintf(fp,"%d %d %.12E %.12E %.12E %d\n",ii,jj,u_x,u_y,pressure,obstacles[ii*params.nx + jj]);
}
}
fclose(fp);
fp = fopen(AVVELSFILE,"w");
if (fp == NULL) {
die("could not open file output file",__LINE__,__FILE__);
}
for (ii=0;ii<params.maxIters;ii++) {
fprintf(fp,"%d:\t%.12E\n", ii, av_vels[ii]);
}
fclose(fp);
}
void die(const char* message, const int line, const char *file)
{
fprintf(stderr, "Error at line %d of file %s:\n", line, file);
fprintf(stderr, "%s\n",message);
fflush(stderr);
exit(EXIT_FAILURE);
}