The Schroedinger equation example..

This example depends on heat-equation.

Table of contents
Introduction About the example The commented program	Results Comments about programming and debugging The plain program

Introduction

This example shows how to use the FFT and iFFT implementations in Ginkgo to solve the non-linear Schrödinger equation with a splitting method.

The non-linear Schrödinger equation (NLS) is given by

$i \partial_t \theta = -\delta \theta + |\theta|^2 \theta$

Here $\theta$ is the wave function of a single particle in two dimensions. Its magnitude $|\theta|^2$ describes the probability distribution of the particle's position.

This equation can be split in to its linear (1) and non-linear (2) part

$\begin{align*} (1) \quad i \partial_t \theta &= -\delta \theta\\ (2) \quad i \partial_t \theta &= |\theta|^2 \theta \end{align*}$

For both of these equations, we can compute exact solutions, assuming periodic boundary conditions and using the Fourier series expansion for (1) and using the fact that $| \theta |^2$ is constant in (2):

$\begin{align*} (\hat 1) \quad \quad \partial_t \hat\theta_k &= -i |k|^2 \theta \\ (2') \quad \partial_t |\theta|^2 &= i |\theta|^2 (\theta - \theta) = 0 \end{align*}$

The exact solutions are then given by

$\begin{align*} (\hat 1) \quad \hat\theta(t) &= e^{-i |k|^2 t} \hat\theta(0)\\ (2') \quad \theta(t) &= e^{-i |\theta|^2 t} \theta(0) \end{align*}$

These partial solutions can be used to approximate a solution to the full NLS by alternating between small time steps for (1) and (2).

For nicer visual results, we add another constant potential term V(x) \theta to the non-linear part, which turns it into the Gross–Pitaevskii equation.

About the example

The commented program

\f}
 
The exact solutions are then given by
 
\f{align*}{
    (\hat 1) \quad \hat\theta(t) &= e^{-i |k|^2 t} \hat\theta(0)\\
    (2') \quad \theta(t) &= e^{-i |\theta|^2 t} \theta(0)
\f}
 
These partial solutions can be used to approximate a solution to the full NLS
by alternating between small time steps for (1) and (2).
 
For nicer visual results, we add another constant potential term V(x) \theta
to the non-linear part, which turns it into the Gross–Pitaevskii equation.
 
*****************************<DESCRIPTION>********************************** /
 
#include <algorithm>
#include <chrono>
#include <fstream>
#include <iostream>
#include <utility>
 
#include <opencv2/core.hpp>
#include <opencv2/videoio.hpp>
 
#include <ginkgo/ginkgo.hpp>

This function implements a simple Ginkgo-themed clamped color mapping for values in the range [0,5].

void set_val(unsigned char* data, double value)

{

RGB values for the 6 colors used for values 0, 1, ..., 5 We will interpolate linearly between these values.

double col_r[] = {255, 221, 129, 201, 249, 255};
double col_g[] = {255, 220, 130, 161, 158, 204};
double col_b[] = {255, 220, 133, 93, 24, 8};
value = std::max(0.0, value);
auto i = std::max(0, std::min(4, int(value)));
auto d = std::max(0.0, std::min(1.0, value - i));

OpenCV uses BGR instead of RGB by default, revert indices

    data[2] = static_cast<unsigned char>(col_r[i + 1] * d + col_r[i] * (1 - d));
    data[1] = static_cast<unsigned char>(col_g[i + 1] * d + col_g[i] * (1 - d));
    data[0] = static_cast<unsigned char>(col_b[i + 1] * d + col_b[i] * (1 - d));
}

Initialize video output with given dimension and FPS (frames per seconds)

std::pair<cv::VideoWriter, cv::Mat> build_output(int n, double fps)
{
    cv::Size videosize{n, n};
    auto output =
        std::make_pair(cv::VideoWriter{}, cv::Mat{videosize, CV_8UC3});
    auto fourcc = cv::VideoWriter::fourcc('a', 'v', 'c', '1');
    output.first.open("nls.mp4", fourcc, fps, videosize);
    return output;
}

Write the current frame to video output using the above color mapping

void output_timestep(std::pair<cv::VideoWriter, cv::Mat>& output, int n,
                     const std::complex<double>* data)
{
    for (int i = 0; i < n; i++) {
        auto row = output.second.ptr(i);
        for (int j = 0; j < n; j++) {
            set_val(&row[3 * j], abs(data[i * n + j]));
        }
    }
    output.first.write(output.second);
}
 
 
int main(int argc, char* argv[])
{
    using vec = gko::matrix::Dense<std::complex<double>>;
    using real_vec = gko::matrix::Dense<double>;
    using fft2 = gko::matrix::Fft2;

Problem parameters: simulation length

const auto t0 = 15.0;

scaling factor for non-linearity

const auto nonlinear_scale = 1.0;

scaling factor for potential

const auto potential_scale = 3.0;

Simulation parameters: time scaling factor

const auto time_scale = 0.25;

number of grid points in each dimension

const auto n = 256;

number of simulation steps per second

const auto steps_per_sec = 1000;

number of video frames per second

const auto fps = 25;

number of grid points

const auto n2 = n * n;

phase difference between neighboring grid points

const auto h = 2.0 * gko::pi<double>() / n;

const auto h2 = h * h;

time step size for the simulation

const auto tau = 1.0 / steps_per_sec;

const auto idx = [&](int i, int j) { return i * n + j; };

create an OpenMP executor

auto exec = gko::OmpExecutor::create();

load initial state vector

std::ifstream initial_stream("data/gko_logo_2d.mtx");
std::ifstream potential_stream("data/gko_text_2d.mtx");
auto amplitude = gko::read<vec>(initial_stream, exec);
auto potential = gko::read<real_vec>(potential_stream, exec);

create vector for frequency space representation

auto frequency = vec::create(exec, amplitude->get_size());

create Fourier matrix

auto fft = fft2::create(exec, n, n);

auto ifft = fft->conj_transpose();

prepare video output

auto output = build_output(n, fps);

time stamp of the last output frame (sentinel value)

double last_t = -t0;

execute splitting method: time step in linear part, then non-linear part

for (double t = 0; t < t0; t += tau) {

if enough time has passed, output the next frame

if (t - last_t > 1.0 / fps) {
    last_t = t;
    std::cout << t << std::endl;
    output_timestep(output, n, amplitude->get_const_values());
}

time step in linear part

fft->apply(amplitude, frequency);
for (int i = 0; i < n; i++) {
    for (int j = 0; j < n; j++) {
        frequency->at(idx(i, j)) *=
            std::polar(1.0, -h2 * (i * i + j * j) * tau * time_scale);

scale by FFT*iFFT normalization factor

        frequency->at(idx(i, j)) *= 1.0 / n2;
    }
}
ifft->apply(frequency, amplitude);

time step in non-linear part

        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                amplitude->at(idx(i, j)) *= std::polar(
                    1.0, -(nonlinear_scale *
                               gko::squared_norm(amplitude->at(idx(i, j))) +
                           potential_scale * potential->at(idx(i, j))) *
                             tau * time_scale);
            }
        }
    }
}

Results

The program will generate a video file named nls.mp4 and output the timestamp of each generated frame.

Comments about programming and debugging

The plain program

 
/*****************************<DESCRIPTION>***********************************
This example shows how to use the FFT and iFFT implementations in Ginkgo
to solve the non-linear Schrödinger equation with a splitting method.
 
The non-linear Schrödinger equation (NLS) is given by
 
@f$
    i \partial_t \theta = -\delta \theta + |\theta|^2 \theta
@f$
 
Here @f$\theta@f$ is the wave function of a single particle in two dimensions.
Its magnitude @f$|\theta|^2@f$ describes the probability distribution of the
particle's position.
 
This equation can be split in to its linear (1) and non-linear (2) part
 
\f{align*}{
    (1) \quad i \partial_t \theta &= -\delta \theta\\
    (2) \quad i \partial_t \theta &= |\theta|^2 \theta
\f}
 
For both of these equations, we can compute exact solutions, assuming periodic
boundary conditions and using the Fourier series expansion for (1) and using the
fact that @f$| \theta |^2@f$ is constant in (2):
 
\f{align*}{
    (\hat 1) \quad \quad \partial_t \hat\theta_k &= -i |k|^2 \theta \\
    (2') \quad \partial_t |\theta|^2 &= i |\theta|^2 (\theta - \theta) = 0
\f}
 
The exact solutions are then given by
 
\f{align*}{
    (\hat 1) \quad \hat\theta(t) &= e^{-i |k|^2 t} \hat\theta(0)\\
    (2') \quad \theta(t) &= e^{-i |\theta|^2 t} \theta(0)
\f}
 
These partial solutions can be used to approximate a solution to the full NLS
by alternating between small time steps for (1) and (2).
 
For nicer visual results, we add another constant potential term V(x) \theta
to the non-linear part, which turns it into the Gross–Pitaevskii equation.
 
*****************************<DESCRIPTION>**********************************/
 
#include <algorithm>
#include <chrono>
#include <fstream>
#include <iostream>
#include <utility>
 
#include <opencv2/core.hpp>
#include <opencv2/videoio.hpp>
 
#include <ginkgo/ginkgo.hpp>
 
 
void set_val(unsigned char* data, double value)
{
    double col_r[] = {255, 221, 129, 201, 249, 255};
    double col_g[] = {255, 220, 130, 161, 158, 204};
    double col_b[] = {255, 220, 133, 93, 24, 8};
    value = std::max(0.0, value);
    auto i = std::max(0, std::min(4, int(value)));
    auto d = std::max(0.0, std::min(1.0, value - i));
    data[2] = static_cast<unsigned char>(col_r[i + 1] * d + col_r[i] * (1 - d));
    data[1] = static_cast<unsigned char>(col_g[i + 1] * d + col_g[i] * (1 - d));
    data[0] = static_cast<unsigned char>(col_b[i + 1] * d + col_b[i] * (1 - d));
}
 
 
std::pair<cv::VideoWriter, cv::Mat> build_output(int n, double fps)
{
    cv::Size videosize{n, n};
    auto output =
        std::make_pair(cv::VideoWriter{}, cv::Mat{videosize, CV_8UC3});
    auto fourcc = cv::VideoWriter::fourcc('a', 'v', 'c', '1');
    output.first.open("nls.mp4", fourcc, fps, videosize);
    return output;
}
 
 
void output_timestep(std::pair<cv::VideoWriter, cv::Mat>& output, int n,
                     const std::complex<double>* data)
{
    for (int i = 0; i < n; i++) {
        auto row = output.second.ptr(i);
        for (int j = 0; j < n; j++) {
            set_val(&row[3 * j], abs(data[i * n + j]));
        }
    }
    output.first.write(output.second);
}
 
 
int main(int argc, char* argv[])
{
    using vec = gko::matrix::Dense<std::complex<double>>;
    using real_vec = gko::matrix::Dense<double>;
    using fft2 = gko::matrix::Fft2;
 
    const auto t0 = 15.0;
    const auto nonlinear_scale = 1.0;
    const auto potential_scale = 3.0;
    const auto time_scale = 0.25;
    const auto n = 256;
    const auto steps_per_sec = 1000;
    const auto fps = 25;
    const auto n2 = n * n;
    const auto h = 2.0 * gko::pi<double>() / n;
    const auto h2 = h * h;
    const auto tau = 1.0 / steps_per_sec;
    const auto idx = [&](int i, int j) { return i * n + j; };
 
    auto exec = gko::OmpExecutor::create();
    std::ifstream initial_stream("data/gko_logo_2d.mtx");
    std::ifstream potential_stream("data/gko_text_2d.mtx");
    auto amplitude = gko::read<vec>(initial_stream, exec);
    auto potential = gko::read<real_vec>(potential_stream, exec);
    auto frequency = vec::create(exec, amplitude->get_size());
    auto fft = fft2::create(exec, n, n);
    auto ifft = fft->conj_transpose();
    auto output = build_output(n, fps);
    double last_t = -t0;
    for (double t = 0; t < t0; t += tau) {
        if (t - last_t > 1.0 / fps) {
            last_t = t;
            std::cout << t << std::endl;
            output_timestep(output, n, amplitude->get_const_values());
        }
        fft->apply(amplitude, frequency);
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                frequency->at(idx(i, j)) *=
                    std::polar(1.0, -h2 * (i * i + j * j) * tau * time_scale);
                frequency->at(idx(i, j)) *= 1.0 / n2;
            }
        }
        ifft->apply(frequency, amplitude);
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                amplitude->at(idx(i, j)) *= std::polar(
                    1.0, -(nonlinear_scale *
                               gko::squared_norm(amplitude->at(idx(i, j))) +
                           potential_scale * potential->at(idx(i, j))) *
                             tau * time_scale);
            }
        }
    }
}