ode_semi_implicit_euler.cpp

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#include <cmath>
#include <ctime>
#include <fstream>
#include <iostream>
#include <valarray>
 
void problem(const double &x, std::valarray<double> *y,
             std::valarray<double> *dy) {
    const double omega = 1.F;             // some const for the problem
    dy[0][0] = y[0][1];                   // x dot
    dy[0][1] = -omega * omega * y[0][0];  // y dot
}
 
void exact_solution(const double &x, std::valarray<double> *y) {
    y[0][0] = std::cos(x);
    y[0][1] = -std::sin(x);
}
 
void semi_implicit_euler_step(const double dx, const double &x,
                              std::valarray<double> *y,
                              std::valarray<double> *dy) {
    problem(x, y, dy);         // update dy once
    y[0][0] += dx * dy[0][0];  // update y0
    problem(x, y, dy);         // update dy once more
 
    dy[0][0] = 0.f;      // ignore y0
    y[0] += dy[0] * dx;  // update remaining using new dy
}
 
double semi_implicit_euler(double dx, double x0, double x_max,
                           std::valarray<double> *y,
                           bool save_to_file = false) {
    std::valarray<double> dy = y[0];
 
    std::ofstream fp;
    if (save_to_file) {
        fp.open("semi_implicit_euler.csv", std::ofstream::out);
        if (!fp.is_open()) {
            std::perror("Error! ");
        }
    }
 
    std::size_t L = y->size();
 
    /* start integration */
    std::clock_t t1 = std::clock();
    double x = x0;
    do {  // iterate for each step of independent variable
        if (save_to_file && fp.is_open()) {
            // write to file
            fp << x << ",";
            for (int i = 0; i < L - 1; i++) {
                fp << y[0][i] << ",";
            }
            fp << y[0][L - 1] << "\n";
        }
 
        semi_implicit_euler_step(dx, x, y, &dy);  // perform integration
        x += dx;                                  // update step
    } while (x <= x_max);  // till upper limit of independent variable
    /* end of integration */
    std::clock_t t2 = std::clock();
 
    if (fp.is_open())
        fp.close();
 
    return static_cast<double>(t2 - t1) / CLOCKS_PER_SEC;
}
 
void save_exact_solution(const double &X0, const double &X_MAX,
                         const double &step_size,
                         const std::valarray<double> &Y0) {
    double x = X0;
    std::valarray<double> y = Y0;
 
    std::ofstream fp("exact.csv", std::ostream::out);
    if (!fp.is_open()) {
        std::perror("Error! ");
        return;
    }
    std::cout << "Finding exact solution\n";
 
    std::clock_t t1 = std::clock();
    do {
        fp << x << ",";
        for (int i = 0; i < y.size() - 1; i++) {
            fp << y[i] << ",";
        }
        fp << y[y.size() - 1] << "\n";
 
        exact_solution(x, &y);
 
        x += step_size;
    } while (x <= X_MAX);
 
    std::clock_t t2 = std::clock();
    double total_time = static_cast<double>(t2 - t1) / CLOCKS_PER_SEC;
    std::cout << "\tTime = " << total_time << " ms\n";
 
    fp.close();
}
 
int main(int argc, char *argv[]) {
    double X0 = 0.f;                       /* initial value of x0 */
    double X_MAX = 10.F;                   /* upper limit of integration */
    std::valarray<double> Y0 = {1.f, 0.f}; /* initial value Y = y(x = x_0) */
    double step_size;
 
    if (argc == 1) {
        std::cout << "\nEnter the step size: ";
        std::cin >> step_size;
    } else {
        // use commandline argument as independent variable step size
        step_size = std::atof(argv[1]);
    }
 
    // get approximate solution
    double total_time = semi_implicit_euler(step_size, X0, X_MAX, &Y0, true);
    std::cout << "\tTime = " << total_time << " ms\n";
 
    /* compute exact solution for comparion */
    save_exact_solution(X0, X_MAX, step_size, Y0);
 
    return 0;
}