TheAlgorithms-C/numerical_methods/ode_forward_euler.c

/**
 * \file
 * \authors [Krishna Vedala](https://github.com/kvedala)
 * \brief Solve a multivariable first order [ordinary differential equation
 * (ODEs)](https://en.wikipedia.org/wiki/Ordinary_differential_equation) using
 * [forward Euler
 * method](https://en.wikipedia.org/wiki/Numerical_methods_for_ordinary_differential_equations#Euler_method)
 *
 * \description
 * The ODE being solved is:
 * \f{eqnarray*}{
 * \dot{u} &=& v\\
 * \dot{v} &=& -\omega^2 u\\
 * \omega &=& 1\\
 * [x_0, u_0, v_0] &=& [0,1,0]\qquad\ldots\text{(initial values)}
 * \f}
 * The exact solution for the above problem is:
 * \f{eqnarray*}{
 * u(x) &=& \cos(x)\\
 * v(x) &=& -\sin(x)\\
 * \f}
 * The computation results are stored to a text file `forward_euler.csv` and the
 * exact soltuion results in `exact.csv` for comparison.
 * <img
 * src="https://raw.githubusercontent.com/kvedala/C/docs/images/numerical_methods/ode_forward_euler.svg"
 * alt="Implementation solution"/>
 */

#include <math.h>
#include <stdio.h>
#include <time.h>

#define order 2 /**< number of dependent variables in ::problem */

/**
 * @brief Problem statement for a system with first-order differential
 * equations. Updates the system differential variables.
 * \note This function can be updated to and ode of any order.
 *
 * @param[in] 		x 		independent variable(s)
 * @param[in,out]	y		dependent variable(s)
 * @param[in,out]	dy	    first-derivative of dependent variable(s)
 */
void problem(double *x, double *y, double *dy)
{
    const double omega = 1.F;      // some const for the problem
    dy[0] = y[1];                  // x dot
    dy[1] = -omega * omega * y[0]; // y dot
}

/**
 * @brief Exact solution of the problem. Used for solution comparison.
 *
 * @param[in] 		x 		independent variable
 * @param[in,out]	y		dependent variable
 */
void exact_solution(double *x, double *y)
{
    y[0] = cos(x[0]);
    y[1] = -sin(x[0]);
}

/**
 * @brief Compute next step approximation using the forward-Euler
 * method. @f[y_{n+1}=y_n + dx\cdot f\left(x_n,y_n\right)@f]
 * @param[in] 		dx	step size
 * @param[in,out] 	x	take \f$x_n\f$ and compute \f$x_{n+1}\f$
 * @param[in,out] 	y	take \f$y_n\f$ and compute \f$y_{n+1}\f$
 * @param[in,out]	dy	compute \f$f\left(x_n,y_n\right)\f$
 */
void forward_euler(double dx, double *x, double *y, double *dy)
{
    int o;
    problem(x, y, dy);
    for (o = 0; o < order; o++)
        y[o] += dx * dy[o];
    *x += dx;
}

/**
    Main Function
*/
int main(int argc, char *argv[])
{
    double X0 = 0.f;          /* initial value of f(x = x0) */
    double Y0[] = {1.f, 0.f}; /* initial value Y = y(x = x_0) */

    double dx, dy[order];
    double x = X0, *y = &(Y0[0]);
    double X_MAX = 10.F; /* upper limit of integration */

    if (argc == 1)
    {
        printf("\nEnter the step size: ");
        scanf("%lg", &dx);
    }
    else
        // use commandline argument as independent variable step size
        dx = atof(argv[1]);

    clock_t t1, t2;
    double total_time;

    FILE *fp = fopen("forward_euler.csv", "w+");
    if (fp == NULL)
    {
        perror("Error! ");
        return -1;
    }
    printf("Computing using 'Forward Euler' algorithm\n");

    /* start integration */
    t1 = clock();
    do // iterate for each step of independent variable
    {
        fprintf(fp, "%.4g,%.4g,%.4g\n", x, y[0], y[1]); // write to file
        forward_euler(dx, &x, y, dy);                   // perform integration
    } while (x <= X_MAX); // till upper limit of independent variable
    /* end of integration */

    t2 = clock();
    fclose(fp);

    total_time = (t2 - t1) / CLOCKS_PER_SEC;
    printf("\tTime taken = %.6g ms\n", total_time);

    /* compute exact solution for comparion */
    fp = fopen("exact.csv", "w+");
    if (fp == NULL)
    {
        perror("Error! ");
        return -1;
    }
    x = X0;
    y = Y0;
    printf("Finding exact solution\n");
    t1 = clock();
    do
    {
        fprintf(fp, "%.4g,%.4g,%.4g\n", x, y[0], y[1]); // write to file
        exact_solution(&x, y);
        x += dx;
    } while (x <= X_MAX);
    t2 = clock();
    total_time = (t2 - t1) / CLOCKS_PER_SEC;
    printf("\tTime = %.6g ms\n", total_time);
    fclose(fp);

    return 0;
}