YouTube-simulations/schrodinger.c
nilsberglund-orleans b1d8db471a
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2021-08-15 11:49:37 +02:00

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C

/*********************************************************************************/
/* */
/* Animation of Schrödinger equation in a planar domain */
/* */
/* N. Berglund, May 2021 */
/* */
/* Feel free to reuse, but if doing so it would be nice to drop a */
/* line to nils.berglund@univ-orleans.fr - Thanks! */
/* */
/* compile with */
/* gcc -o schrodinger schrodinger.c */
/* -L/usr/X11R6/lib -ltiff -lm -lGL -lGLU -lX11 -lXmu -lglut -O3 -fopenmp */
/* */
/* To make a video, set MOVIE to 1 and create subfolder tif_schrod */
/* It may be possible to increase parameter PAUSE */
/* */
/* create movie using */
/* ffmpeg -i wave.%05d.tif -vcodec libx264 wave.mp4 */
/* */
/*********************************************************************************/
/*********************************************************************************/
/* */
/* NB: The algorithm used to simulate the wave equation is highly paralellizable */
/* One could make it much faster by using a GPU */
/* */
/*********************************************************************************/
#include <math.h>
#include <string.h>
#include <GL/glut.h>
#include <GL/glu.h>
#include <unistd.h>
#include <sys/types.h>
#include <tiffio.h> /* Sam Leffler's libtiff library. */
#include <omp.h>
#define MOVIE 0 /* set to 1 to generate movie */
/* General geometrical parameters */
#define WINWIDTH 1280 /* window width */
#define WINHEIGHT 720 /* window height */
#define NX 1280 /* number of grid points on x axis */
#define NY 720 /* number of grid points on y axis */
// #define NX 640 /* number of grid points on x axis */
// #define NY 360 /* number of grid points on y axis */
/* setting NX to WINWIDTH and NY to WINHEIGHT increases resolution */
/* but will multiply run time by 4 */
#define XMIN -2.0
#define XMAX 2.0 /* x interval */
#define YMIN -1.125
#define YMAX 1.125 /* y interval for 9/16 aspect ratio */
#define JULIA_SCALE 1.0 /* scaling for Julia sets */
/* Choice of the billiard table, see list in global_pdes.c */
#define B_DOMAIN 9 /* choice of domain shape */
#define CIRCLE_PATTERN 0 /* pattern of circles, see list in global_pdes.c */
#define P_PERCOL 0.25 /* probability of having a circle in C_RAND_PERCOL arrangement */
#define NPOISSON 300 /* number of points for Poisson C_RAND_POISSON arrangement */
#define LAMBDA 0.3 /* parameter controlling the dimensions of domain */
#define MU 0.05 /* parameter controlling the dimensions of domain */
#define NPOLY 6 /* number of sides of polygon */
#define APOLY 1.0 /* angle by which to turn polygon, in units of Pi/2 */
#define MDEPTH 3 /* depth of computation of Menger gasket */
#define MRATIO 3 /* ratio defining Menger gasket */
#define MANDELLEVEL 1000 /* iteration level for Mandelbrot set */
#define MANDELLIMIT 10.0 /* limit value for approximation of Mandelbrot set */
#define FOCI 1 /* set to 1 to draw focal points of ellipse */
#define NGRIDX 15 /* number of grid point for grid of disks */
#define NGRIDY 20 /* number of grid point for grid of disks */
/* You can add more billiard tables by adapting the functions */
/* xy_in_billiard and draw_billiard in sub_wave.c */
/* Physical patameters of wave equation */
// #define DT 0.00000005
#define DT 0.000000005
// #define DT 0.000000005
#define HBAR 1.0
/* Boundary conditions, see list in global_pdes.c */
#define B_COND 1
/* Parameters for length and speed of simulation */
#define NSTEPS 200 /* number of frames of movie */
#define NVID 1200 /* number of iterations between images displayed on screen */
#define NSEG 100 /* number of segments of boundary */
#define BOUNDARY_WIDTH 2 /* width of billiard boundary */
#define PAUSE 1000 /* number of frames after which to pause */
#define PSLEEP 1 /* sleep time during pause */
#define SLEEP1 1 /* initial sleeping time */
#define SLEEP2 1 /* final sleeping time */
/* For debugging purposes only */
#define FLOOR 0 /* set to 1 to limit wave amplitude to VMAX */
#define VMAX 10.0 /* max value of wave amplitude */
/* Plot type, see list in global_pdes.c */
#define PLOT 10
/* Color schemes, see list in global_pdes.c */
#define BLACK 1 /* black background */
#define COLOR_SCHEME 1 /* choice of color scheme */
#define SCALE 1 /* set to 1 to adjust color scheme to variance of field */
#define SLOPE 1.0 /* sensitivity of color on wave amplitude */
#define ATTENUATION 0.0 /* exponential attenuation coefficient of contrast with time */
#define COLORHUE 260 /* initial hue of water color for scheme C_LUM */
#define COLORDRIFT 0.0 /* how much the color hue drifts during the whole simulation */
#define LUMMEAN 0.5 /* amplitude of luminosity variation for scheme C_LUM */
#define LUMAMP 0.3 /* amplitude of luminosity variation for scheme C_LUM */
#define HUEMEAN 150.0 /* mean value of hue for color scheme C_HUE */
#define HUEAMP -150.0 /* amplitude of variation of hue for color scheme C_HUE */
#include "global_pdes.c"
#include "sub_wave.c"
double courant2; /* Courant parameter squared */
double dx2; /* spatial step size squared */
double intstep; /* integration step */
double intstep1; /* integration step used in absorbing boundary conditions */
void init_coherent_state(double x, double y, double px, double py, double scalex, double *phi[NX],
double *psi[NX], short int *xy_in[NX])
/* initialise field with coherent state of position (x,y) and momentum (px, py) */
/* phi is real part, psi is imaginary part */
{
int i, j;
double xy[2], dist2, module, phase, scale2;
scale2 = scalex*scalex;
for (i=0; i<NX; i++)
for (j=0; j<NY; j++)
{
ij_to_xy(i, j, xy);
xy_in[i][j] = xy_in_billiard(xy[0],xy[1]);
if (xy_in[i][j])
{
dist2 = (xy[0]-x)*(xy[0]-x) + (xy[1]-y)*(xy[1]-y);
module = exp(-dist2/scale2);
if (module < 1.0e-15) module = 1.0e-15;
phase = (px*(xy[0]-x) + py*(xy[1]-y))/scalex;
phi[i][j] = module*cos(phase);
psi[i][j] = module*sin(phase);
}
else
{
phi[i][j] = 0.0;
psi[i][j] = 0.0;
}
}
}
/*********************/
/* animation part */
/*********************/
void schrodinger_color_scheme(double phi, double psi, double scale, int time, double rgb[3])
// double phi, psi, scale, rgb[3];
// int time;
{
double phase, amp, lum;
if (PLOT == P_MODULE)
color_scheme(COLOR_SCHEME, 2.0*module2(phi, psi)-1.0, scale, time, rgb);
else if (PLOT == P_PHASE)
{
amp = module2(phi,psi);
// if (amp < 1.0e-10) amp = 1.0e-10;
phase = argument(phi/amp, psi/amp);
if (phase < 0.0) phase += DPI;
lum = (color_amplitude(amp, scale, time))*0.5;
if (lum < 0.0) lum = 0.0;
hsl_to_rgb(phase*360.0/DPI, 0.9, lum, rgb);
}
else if (PLOT == P_REAL) color_scheme(COLOR_SCHEME, phi, scale, time, rgb);
else if (PLOT == P_IMAGINARY) color_scheme(COLOR_SCHEME, psi, scale, time, rgb);
}
void draw_wave(double *phi[NX], double *psi[NX], short int *xy_in[NX], double scale, int time)
/* draw the field */
{
int i, j;
double rgb[3], xy[2], x1, y1, x2, y2, amp, phase;
glBegin(GL_QUADS);
for (i=0; i<NX; i++)
for (j=0; j<NY; j++)
{
if (xy_in[i][j])
{
schrodinger_color_scheme(phi[i][j],psi[i][j], scale, time, rgb);
glColor3f(rgb[0], rgb[1], rgb[2]);
glVertex2i(i, j);
glVertex2i(i+1, j);
glVertex2i(i+1, j+1);
glVertex2i(i, j+1);
}
}
glEnd ();
}
void evolve_wave_half(double *phi_in[NX], double *psi_in[NX], double *phi_out[NX], double *psi_out[NX],
short int *xy_in[NX])
// void evolve_wave_half(phi_in, psi_in, phi_out, psi_out, xy_in)
// /* time step of field evolution */
// /* phi is real part, psi is imaginary part */
// double *phi_in[NX], *psi_in[NX], *phi_out[NX], *psi_out[NX]; short int *xy_in[NX];
{
int i, j, iplus, iminus, jplus, jminus;
double delta1, delta2, x, y;
#pragma omp parallel for private(i,j,iplus,iminus,jplus,jminus,delta1,delta2,x,y)
for (i=0; i<NX; i++){
for (j=0; j<NY; j++){
if (xy_in[i][j]){
/* discretized Laplacian depending on boundary conditions */
if ((B_COND == BC_DIRICHLET)||(B_COND == BC_ABSORBING))
{
iplus = (i+1); if (iplus == NX) iplus = NX-1;
iminus = (i-1); if (iminus == -1) iminus = 0;
jplus = (j+1); if (jplus == NY) jplus = NY-1;
jminus = (j-1); if (jminus == -1) jminus = 0;
}
else if (B_COND == BC_PERIODIC)
{
iplus = (i+1) % NX;
iminus = (i-1) % NX;
if (iminus < 0) iminus += NX;
jplus = (j+1) % NY;
jminus = (j-1) % NY;
if (jminus < 0) jminus += NY;
}
delta1 = phi_in[iplus][j] + phi_in[iminus][j] + phi_in[i][jplus] + phi_in[i][jminus] - 4.0*phi_in[i][j];
delta2 = psi_in[iplus][j] + psi_in[iminus][j] + psi_in[i][jplus] + psi_in[i][jminus] - 4.0*psi_in[i][j];
x = phi_in[i][j];
y = psi_in[i][j];
/* evolve phi and psi */
if (B_COND != BC_ABSORBING)
{
phi_out[i][j] = x - intstep*delta2;
psi_out[i][j] = y + intstep*delta1;
}
else /* case of absorbing b.c. - this is only an approximation of correct way of implementing */
{
/* in the bulk */
if ((i>0)&&(i<NX-1)&&(j>0)&&(j<NY-1))
{
phi_out[i][j] = x - intstep*delta2;
psi_out[i][j] = y + intstep*delta1;
}
/* right border */
else if (i==NX-1)
{
phi_out[i][j] = x - intstep1*(y - psi_in[i-1][j]);
psi_out[i][j] = y + intstep1*(x - phi_in[i-1][j]);
}
/* upper border */
else if (j==NY-1)
{
phi_out[i][j] = x - intstep1*(y - psi_in[i][j-1]);
psi_out[i][j] = y + intstep1*(x - phi_in[i][j-1]);
}
/* left border */
else if (i==0)
{
phi_out[i][j] = x - intstep1*(y - psi_in[1][j]);
psi_out[i][j] = y + intstep1*(x - phi_in[1][j]);
}
/* lower border */
else if (j==0)
{
phi_out[i][j] = x - intstep1*(y - psi_in[i][1]);
psi_out[i][j] = y + intstep1*(x - phi_in[i][1]);
}
}
if (FLOOR)
{
if (phi_out[i][j] > VMAX) phi_out[i][j] = VMAX;
if (phi_out[i][j] < -VMAX) phi_out[i][j] = -VMAX;
if (psi_out[i][j] > VMAX) psi_out[i][j] = VMAX;
if (psi_out[i][j] < -VMAX) psi_out[i][j] = -VMAX;
}
}
}
}
// printf("phi(0,0) = %.3lg, psi(0,0) = %.3lg\n", phi[NX/2][NY/2], psi[NX/2][NY/2]);
}
void evolve_wave(double *phi[NX], double *psi[NX], double *phi_tmp[NX], double *psi_tmp[NX], short int *xy_in[NX])
/* time step of field evolution */
/* phi is real part, psi is imaginary part */
{
evolve_wave_half(phi, psi, phi_tmp, psi_tmp, xy_in);
evolve_wave_half(phi_tmp, psi_tmp, phi, psi, xy_in);
}
double compute_variance(double *phi[NX], double *psi[NX], short int *xy_in[NX])
// double compute_variance(phi, psi, xy_in)
/* compute the variance (total probability) of the field */
// double *phi[NX], *psi[NX]; short int * xy_in[NX];
{
int i, j, n = 0;
double variance = 0.0;
for (i=1; i<NX; i++)
for (j=1; j<NY; j++)
{
if (xy_in[i][j])
{
n++;
variance += phi[i][j]*phi[i][j] + psi[i][j]*psi[i][j];
}
}
if (n==0) n=1;
return(variance/(double)n);
}
void renormalise_field(double *phi[NX], double *psi[NX], short int *xy_in[NX], double variance)
/* renormalise variance of field */
{
int i, j;
double stdv;
stdv = sqrt(variance);
for (i=1; i<NX; i++)
for (j=1; j<NY; j++)
{
if (xy_in[i][j])
{
phi[i][j] = phi[i][j]/stdv;
psi[i][j] = psi[i][j]/stdv;
}
}
}
void animation()
{
double time, scale, dx, var;
double *phi[NX], *psi[NX], *phi_tmp[NX], *psi_tmp[NX];
short int *xy_in[NX];
int i, j, s;
/* Since NX and NY are big, it seemed wiser to use some memory allocation here */
for (i=0; i<NX; i++)
{
phi[i] = (double *)malloc(NY*sizeof(double));
psi[i] = (double *)malloc(NY*sizeof(double));
phi_tmp[i] = (double *)malloc(NY*sizeof(double));
psi_tmp[i] = (double *)malloc(NY*sizeof(double));
xy_in[i] = (short int *)malloc(NY*sizeof(short int));
}
dx = (XMAX-XMIN)/((double)NX);
intstep = DT/(dx*dx*HBAR);
intstep1 = DT/(dx*HBAR);
printf("Integration step %.3lg\n", intstep);
/* initialize wave wave function */
init_coherent_state(-1.2, 0.0, 20.0, 0.0, 0.25, phi, psi, xy_in);
// init_coherent_state(0.0, 0.0, 0.0, 5.0, 0.03, phi, psi, xy_in);
// init_coherent_state(-0.5, 0.0, 1.0, 1.0, 0.05, phi, psi, xy_in);
if (SCALE)
{
var = compute_variance(phi,psi, xy_in);
scale = sqrt(1.0 + var);
renormalise_field(phi, psi, xy_in, var);
}
blank();
glColor3f(0.0, 0.0, 0.0);
glutSwapBuffers();
sleep(SLEEP1);
for (i=0; i<=NSTEPS; i++)
{
/* compute the variance of the field to adjust color scheme */
/* the color depends on the field divided by sqrt(1 + variance) */
if (SCALE)
{
var = compute_variance(phi,psi, xy_in);
scale = sqrt(1.0 + var);
// printf("Norm: %5lg\t Scaling factor: %5lg\n", var, scale);
renormalise_field(phi, psi, xy_in, var);
}
else scale = 1.0;
draw_wave(phi, psi, xy_in, scale, i);
// printf("Wave drawn\n");
for (j=0; j<NVID; j++) evolve_wave(phi, psi, phi_tmp, psi_tmp, xy_in);
draw_billiard();
glutSwapBuffers();
if (MOVIE)
{
save_frame();
/* it seems that saving too many files too fast can cause trouble with the file system */
/* so this is to make a pause from time to time - parameter PAUSE may need adjusting */
if (i % PAUSE == PAUSE - 1)
{
printf("Making a short pause\n");
sleep(PSLEEP);
s = system("mv wave*.tif tif_schrod/");
}
}
}
// if (MOVIE)
// {
// for (i=0; i<20; i++) save_frame();
// s = system("mv wave*.tif tif_schrod/");
// }
for (i=0; i<NX; i++)
{
free(phi[i]);
free(psi[i]);
free(phi_tmp[i]);
free(psi_tmp[i]);
free(xy_in[i]);
}
}
void display(void)
{
glPushMatrix();
blank();
glutSwapBuffers();
blank();
glutSwapBuffers();
animation();
sleep(SLEEP2);
glPopMatrix();
glutDestroyWindow(glutGetWindow());
}
int main(int argc, char** argv)
{
glutInit(&argc, argv);
glutInitDisplayMode(GLUT_RGB | GLUT_DOUBLE | GLUT_DEPTH);
glutInitWindowSize(WINWIDTH,WINHEIGHT);
glutCreateWindow("Schrodinger equation in a planar domain");
init();
glutDisplayFunc(display);
glutMainLoop();
return 0;
}