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