Runge-Kutta Methods – C PROGRAM

Runge-Kutta Method is a numerical technique to find the solution of ordinary differential equations.

The second-order Runge-Kutta method uses the following formula:
k_1=hf(x_i,y_i)
k_2=hf(x_i+h/2, y_i+k_1/2)
y_{i+1}=y_{i} + k_2 + O(h^3)

The fourth-order Runge-Kutta method uses the following formula:
k_1=hf(x_i,y_i)
k_2=hf(x_i+h/2, _i+k_1/2)
k_3=hf(x_i+h/2,y_i+k_2/2)
k_4=hf(x_i+h, y_i+k_3)

The program for the second-order Runge-Kutta Method is shown below:

PROGRAM(RK II ORDER:

/**************************************
*********RUNGE-KUTTA METHOD(1)*********
**************************************/
#include<stdio.h>
#include<math.h>
/*Define the RHS of the first order differential equation here(Ex: dy/dx=f(x,y))  */
double f(double x, double y){
	//return 2-exp(-4*x)-2*y;	
	//return x+y;
	return x;
}
main(){
	int i;
	double x,y,x0,y0,h,k1,k2;
	printf("Enter the initial condition for y: ");
	scanf("%lf",&y0);
	printf("Enter the initial condition for x: ");
	scanf("%lf",&x0);
	printf("Enter the value of x for which y is required: ");
	scanf("%lf",&x);
	printf("Enter the step-width h: ");
	scanf("%lf",&h);
	printf("x\t\ty\t\ty'\t\tk1\t\tk2\n");
	printf("__________________________________________________________________________\n");
	//Begin Runge-Kutta Routine
	while((x-x0)>0.0000000001){
		k1=h*f(x0,y0);
		k2=h*f(x0+h/2.0,y0+k1/2.0);
		y=y0+k2;
		printf("%lf\t%lf\t%lf\t%lf\t%lf\n",x0,y0,f(x0,y0),k1,k2);
		y0=y;
		x0=x0+h;
	}
	printf("%lf\t%lf\n",x0,y0);
	printf("__________________________________________________________________________\n");
	printf("The value of y is %lf\n\n",y);
}

OUTPUT:

The program for the fourth-order Runge-Kutta Method is shown below:

PROGRAM(RK 4th ODER):

/**************************************
*********RUNGE-KUTTA METHOD(2)*********
**************************************/
#include<stdio.h>
#include<math.h>
/*Define the RHS of the first order differential equation here(Ex: dy/dx=f(x,y))  */
double f(double x, double y){
	//return 2-exp(-4*x)-2*y;	
	//return x+y;
	return x;
}
main(){
	int i;
	double x,y,x0,y0,h,k1,k2,k3,k4;
	printf("Enter the initial condition for y: ");
	scanf("%lf",&y0);
	printf("Enter the initial condition for x: ");
	scanf("%lf",&x0);
	printf("Enter the value of x for which y is required: ");
	scanf("%lf",&x);
	printf("Enter the step-width h: ");
	scanf("%lf",&h);
	printf("x\t\ty\t\tk1\t\tk2\t\tk3\t\tk4\t\tk_avg\n");
	printf("__________________________________________________________________________________________________________\n");
	//Begin Runge-Kutta Routine
	while((x-x0)>0.0000000001){
		k1=h*f(x0,y0);
		k2=h*f(x0+h/2.0,y0+k1/2.0);
		k3=h*f(x0+h/2.0,y0+k2/2.0);
		k4=h*f(x0+h,y0+k3);
		printf("%lf\t%lf\t%lf\t%lf\t%lf\t%lf\t%lf\n",x0,y0,k1,k2,k3,k4,1/6.0*(k1+2*k2+2*k3+k4));
		y=y0+1/6.0*(k1+2*k2+2*k3+k4);
		y0=y;
		x0=x0+h;
	}
	printf("%lf\t%lf\n",x0,y0);
	printf("____________________________________________________________________________________________________________\n");
	printf("The value of y is %lf\n\n",y);
}

OUTPUT:

I'm a physicist specializing in theoretical, computational and experimental condensed matter physics. I like to develop Physics related apps and softwares from time to time. Can code in most of the popular languages. Like to share my knowledge in Physics and applications using this Blog and a YouTube channel.



Leave a Reply

Your email address will not be published. Required fields are marked *