The heat-equation program

The heat equation example..

This example depends on simple-solver, three-pt-stencil-solver.

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

Introduction

This example solves a 2D heat conduction equation

with Dirichlet boundary conditions and given initial condition and constant-in-time source function f.

The partial differential equation (PDE) is solved with a finite difference spatial discretization on an equidistant grid: For n grid points, and grid distance we write

We then build an implicit Euler integrator by discretizing with time step

and solve the resulting linear system for using Ginkgo's CG solver preconditioned with an incomplete Cholesky factorization for each time step, occasionally writing the resulting grid values into a video file using OpenCV and a custom color mapping.

The intention of this example is to provide a mini-app showing matrix assembly, vector initialization, solver setup and the use of Ginkgo in a more complex setting.

About the example

The commented program

The intention of this example is to provide a mini-app showing matrix assembly,
vector initialization, solver setup and the use of Ginkgo in a more complex
setting.
*****************************<DESCRIPTION>********************************** /
 
#include <chrono>
#include <fstream>
#include <iostream>
 
#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("heat.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 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], data[i * n + j]);
        }
    }
    output.first.write(output.second);
}
 
 
int main(int argc, char* argv[])
{
    using mtx = gko::matrix::Csr<>;
    using vec = gko::matrix::Dense<>;

Problem parameters: simulation length

auto t0 = 5.0;

diffusion factor

auto diffusion = 0.0005;

scaling factor for heat source

auto source_scale = 2.5;

Simulation parameters: inner grid points per discretization direction

auto n = 256;

number of simulation steps per second

auto steps_per_sec = 500;

number of video frames per second

auto fps = 25;

number of grid points

auto n2 = n * n;

grid point distance (ignoring boundary points)

auto h = 1.0 / (n + 1);
auto h2 = h * h;

time step size for the simulation

auto tau = 1.0 / steps_per_sec;

create a CUDA executor with an associated OpenMP host executor

auto exec = gko::CudaExecutor::create(0, gko::OmpExecutor::create());

load heat source and initial state vectors

std::ifstream initial_stream("data/gko_logo_2d.mtx");
std::ifstream source_stream("data/gko_text_2d.mtx");
auto source = gko::read<vec>(source_stream, exec);
auto in_vector = gko::read<vec>(initial_stream, exec);

create output vector with initial guess for

auto out_vector = in_vector->clone();

create scalar for source update

auto tau_source_scalar = gko::initialize<vec>({source_scale * tau}, exec);

create stencil matrix as shared_ptr for solver

auto stencil_matrix = gko::share(mtx::create(exec));

assemble matrix

gko::matrix_data<> mtx_data{gko::dim<2>(n2, n2)};
for (int i = 0; i < n; i++) {
    for (int j = 0; j < n; j++) {
        auto c = i * n + j;
        auto c_val = diffusion * tau * 4.0 / h2 + 1.0;
        auto off_val = -diffusion * tau / h2;

for each grid point: insert 5 stencil points with Dirichlet boundary conditions, i.e. with zero boundary value

        if (i > 0) {
            mtx_data.nonzeros.emplace_back(c, c - n, off_val);
        }
        if (j > 0) {
            mtx_data.nonzeros.emplace_back(c, c - 1, off_val);
        }
        mtx_data.nonzeros.emplace_back(c, c, c_val);
        if (j < n - 1) {
            mtx_data.nonzeros.emplace_back(c, c + 1, off_val);
        }
        if (i < n - 1) {
            mtx_data.nonzeros.emplace_back(c, c + n, off_val);
        }
    }
}
stencil_matrix->read(mtx_data);

prepare video output

auto output = build_output(n, fps);

build CG solver on stencil with incomplete Cholesky preconditioner stopping at 1e-10 relative accuracy

auto solver =
    gko::solver::Cg<>::build()
        .with_preconditioner(gko::preconditioner::Ic<>::build())
        .with_criteria(gko::stop::ResidualNorm<>::build()
                           .with_baseline(gko::stop::mode::rhs_norm)
                           .with_reduction_factor(1e-10))
        .on(exec)
        ->generate(stencil_matrix);

time stamp of the last output frame (initialized to a sentinel value)

double last_t = -t0;

execute implicit Euler method: for each timestep, solve stencil system

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

if enough time has passed, output the next video frame

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

add heat source contribution

in_vector->add_scaled(tau_source_scalar, source);

execute Euler step

solver->apply(in_vector, out_vector);

swap input and output

        std::swap(in_vector, out_vector);
    }
}

Results

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

Comments about programming and debugging

The plain program

 
/*****************************<DESCRIPTION>***********************************
This example solves a 2D heat conduction equation
 
    u : [0, d]^2 \rightarrow R\\
    \partial_t u = \delta u + f
 
with Dirichlet boundary conditions and given initial condition and
constant-in-time source function f.
 
The partial differential equation (PDE) is solved with a finite difference
spatial discretization on an equidistant grid: For `n` grid points,
and grid distance @f$h = 1/n@f$ we write
 
    u_{i,j}' = \alpha (u_{i-1,j} + u_{i+1,j} + u_{i,j-1} + u_{i,j+1}
                   - 4 u_{i,j}) / h^2
               + f_{i,j}
 
We then build an implicit Euler integrator by discretizing with time step @f$\tau@f$
 
    (u_{i,j}^{k+1} - u_{i,j}^k) / \tau =
    \alpha (u_{i-1,j}^{k+1} - u_{i+1,j}^{k+1}
          + u_{i,j-1}^{k+1} - u_{i,j+1}^{k+1} - 4 u_{i,j}^{k+1}) / h^2
    + f_{i,j}
 
and solve the resulting linear system for @f$ u_{\cdot}^{k+1}@f$ using Ginkgo's CG
solver preconditioned with an incomplete Cholesky factorization for each time
step, occasionally writing the resulting grid values into a video file using
OpenCV and a custom color mapping.
 
The intention of this example is to provide a mini-app showing matrix assembly,
vector initialization, solver setup and the use of Ginkgo in a more complex
setting.
*****************************<DESCRIPTION>**********************************/
 
#include <chrono>
#include <fstream>
#include <iostream>
 
#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("heat.mp4", fourcc, fps, videosize);
    return output;
}
 
 
void output_timestep(std::pair<cv::VideoWriter, cv::Mat>& output, int n,
                     const 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], data[i * n + j]);
        }
    }
    output.first.write(output.second);
}
 
 
int main(int argc, char* argv[])
{
    using mtx = gko::matrix::Csr<>;
    using vec = gko::matrix::Dense<>;
 
    auto t0 = 5.0;
    auto diffusion = 0.0005;
    auto source_scale = 2.5;
    auto n = 256;
    auto steps_per_sec = 500;
    auto fps = 25;
    auto n2 = n * n;
    auto h = 1.0 / (n + 1);
    auto h2 = h * h;
    auto tau = 1.0 / steps_per_sec;
 
    auto exec = gko::CudaExecutor::create(0, gko::OmpExecutor::create());
    std::ifstream initial_stream("data/gko_logo_2d.mtx");
    std::ifstream source_stream("data/gko_text_2d.mtx");
    auto source = gko::read<vec>(source_stream, exec);
    auto in_vector = gko::read<vec>(initial_stream, exec);
    auto out_vector = in_vector->clone();
    auto tau_source_scalar = gko::initialize<vec>({source_scale * tau}, exec);
    auto stencil_matrix = gko::share(mtx::create(exec));
    gko::matrix_data<> mtx_data{gko::dim<2>(n2, n2)};
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            auto c = i * n + j;
            auto c_val = diffusion * tau * 4.0 / h2 + 1.0;
            auto off_val = -diffusion * tau / h2;
            if (i > 0) {
                mtx_data.nonzeros.emplace_back(c, c - n, off_val);
            }
            if (j > 0) {
                mtx_data.nonzeros.emplace_back(c, c - 1, off_val);
            }
            mtx_data.nonzeros.emplace_back(c, c, c_val);
            if (j < n - 1) {
                mtx_data.nonzeros.emplace_back(c, c + 1, off_val);
            }
            if (i < n - 1) {
                mtx_data.nonzeros.emplace_back(c, c + n, off_val);
            }
        }
    }
    stencil_matrix->read(mtx_data);
    auto output = build_output(n, fps);
    auto solver =
        gko::solver::Cg<>::build()
            .with_preconditioner(gko::preconditioner::Ic<>::build())
            .with_criteria(gko::stop::ResidualNorm<>::build()
                               .with_baseline(gko::stop::mode::rhs_norm)
                               .with_reduction_factor(1e-10))
            .on(exec)
            ->generate(stencil_matrix);
    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,
                gko::make_temporary_clone(exec->get_master(), in_vector)
                    ->get_const_values());
        }
        in_vector->add_scaled(tau_source_scalar, source);
        solver->apply(in_vector, out_vector);
        std::swap(in_vector, out_vector);
    }
}