ode_semi_implicit_euler.c File Reference

Solve a multivariable first order ordinary differential equation (ODEs) using semi implicit Euler method More...

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

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Macros
#define	order   2
	number of dependent variables in problem

Functions
void	problem (const double *x, double *y, double *dy)
	Problem statement for a system with first-order differential equations. More...
void	exact_solution (const double *x, double *y)
	Exact solution of the problem. More...
void	semi_implicit_euler_step (double dx, double *x, double *y, double *dy)
	Compute next step approximation using the semi-implicit-Euler method. More...
double	semi_implicit_euler (double dx, double x0, double x_max, double *y, char save_to_file)
	Compute approximation using the semi-implicit-Euler method in the given limits. More...
int	main (int argc, char *argv[])
	Main Function. More...

Detailed Description

Solve a multivariable first order ordinary differential equation (ODEs) using semi implicit Euler method

Authors

Krishna Vedala

The ODE being solved is:

\begin{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)} \end{eqnarray*}

The exact solution for the above problem is:

\begin{eqnarray*} u(x) &=& \cos(x)\\ v(x) &=& -\sin(x)\\ \end{eqnarray*}

The computation results are stored to a text file semi_implicit_euler.csv and the exact soltuion results in exact.csv for comparison.

To implement Van der Pol oscillator, change the problem function to:

const double mu = 2.0;
dy[0] = y[1];
dy[1] = mu * (1.f - y[0] * y[0]) * y[1] - y[0];

See also

ode_forward_euler.c, ode_midpoint_euler.c

Function Documentation

exact_solution()

void exact_solution	(	const double *	x,
		double *	y
	)

Exact solution of the problem.

Used for solution comparison.

Parameters

[in]	x	independent variable
[in,out]	y	dependent variable

{
    y[0] = cos(x[0]);
    y[1] = -sin(x[0]);
}

main()

int main	(	int	argc,
		char *	argv[]
	)

Main Function.

{
    double X0 = 0.f;          /* initial value of x0 */
    double X_MAX = 10.F;      /* upper limit of integration */
    double Y0[] = {1.f, 0.f}; /* initial value Y = y(x = x_0) */
    double step_size;
 
    if (argc == 1)
    {
        printf("\nEnter the step size: ");
        scanf("%lg", &step_size);
    }
    else
        // use commandline argument as independent variable step size
        step_size = atof(argv[1]);
 
    // get approximate solution
    double total_time = semi_implicit_euler(step_size, X0, X_MAX, Y0, 1);
    printf("\tTime = %.6g ms\n", total_time);
 
    /* compute exact solution for comparion */
    FILE *fp = fopen("exact.csv", "w+");
    if (fp == NULL)
    {
        perror("Error! ");
        return -1;
    }
    double x = X0;
    double *y = &(Y0[0]);
    printf("Finding exact solution\n");
    clock_t t1 = clock();
 
    do
    {
        fprintf(fp, "%.4g,%.4g,%.4g\n", x, y[0], y[1]);  // write to file
        exact_solution(&x, y);
        x += step_size;
    } while (x <= X_MAX);
 
    clock_t t2 = clock();
    total_time = (t2 - t1) / CLOCKS_PER_SEC;
    printf("\tTime = %.6g ms\n", total_time);
    fclose(fp);
 
    return 0;
}

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problem()

void problem	(	const double *	x,
		double *	y,
		double *	dy
	)

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.

Parameters

[in]	x	independent variable(s)
[in,out]	y	dependent variable(s)
[in,out]	dy	first-derivative of dependent variable(s)

{
    const double omega = 1.F;       // some const for the problem
    dy[0] = y[1];                   // x dot
    dy[1] = -omega * omega * y[0];  // y dot
}

semi_implicit_euler()

double semi_implicit_euler	(	double	dx,
		double	x0,
		double	x_max,
		double *	y,
		char	save_to_file
	)

Compute approximation using the semi-implicit-Euler method in the given limits.

Parameters

[in]	dx	step size
[in]	x0	initial value of independent variable
[in]	x_max	final value of independent variable
[in,out]	y	take \(y_n\) and compute \(y_{n+1}\)
[in]	save_to_file	flag to save results to a CSV file (1) or not (0)

Returns

time taken for computation in seconds

{
    double dy[order];
 
    FILE *fp = NULL;
    if (save_to_file)
    {
        fp = fopen("semi_implicit_euler.csv", "w+");
        if (fp == NULL)
        {
            perror("Error! ");
            return -1;
        }
    }
 
    /* start integration */
    clock_t t1 = clock();
    double x = x0;
    do  // iterate for each step of independent variable
    {
        if (save_to_file && fp)
            fprintf(fp, "%.4g,%.4g,%.4g\n", x, y[0], y[1]);  // write to file
        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 */
    clock_t t2 = clock();
 
    if (save_to_file && fp)
        fclose(fp);
 
    return (double)(t2 - t1) / CLOCKS_PER_SEC;
}

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semi_implicit_euler_step()

void semi_implicit_euler_step	(	double	dx,
		double *	x,
		double *	y,
		double *	dy
	)

Compute next step approximation using the semi-implicit-Euler method.

Parameters

[in]	dx	step size
[in,out]	x	take \(x_n\) and compute \(x_{n+1}\)
[in,out]	y	take \(y_n\) and compute \(y_{n+1}\)
[in,out]	dy	compute \(y_n+\frac{1}{2}dx\,f\left(x_n,y_n\right)\)

{
    int o;
 
    problem(x, y, dy);   // update dy once
    y[0] += dx * dy[0];  // update y0
 
    problem(x, y, dy);  // update dy once more
 
    for (o = 1; o < order; o++)
        y[o] += dx * dy[o];  // update remaining using new dy
    *x += dx;
}

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