Runge Kutta Method Formula and C Program

Runge Kutta Method

A Runge Kutta Method of numerically integrating ordinary differential equations by using a trial step at the midpoint of an interval to cancel out lower-order error terms. The second-order formula is

Each of these slope estimates can be described verbally.

• k₁ is the slope at the beginning of the time step.
• If we use the slope k₁ to step halfway through the time step, then k₂ is an estimate of the slope at the midpoint. This is the same as the slope, k₂, from the second-order midpoint method. This slope proved to be more accurate than k₁ for making new approximations for y(t).
• If we use the slope k₂ to step halfway through the time step, then k₃ is another estimate of the slope at the midpoint.
• Finally, we use the slope, k₃, to step all the way across the time step (to t₀+h), and k₄ is an estimate of the slope at the endpoint.

C Code For 4th Order Runge Kutta Method

#include<stdio.h>
#include<math.h>
#include<conio.h>
#define f(x,y) 1-2*(x)*(x)*(y)

int main(){
 float x,xp,x0,y0,y,h,m1,m2,m3,m4;
 printf("\n runge-Kutta Forth order\n");
 printf("\nEnter initial values of x and y\n");
 scanf("%f%f",&x0,&y0);
 printf("\nEnter x at which function to be evaluated\n");
 scanf("%f",&xp);
 printf("\nEnter the step size \n");
 scanf("%f",&h);
 y=y0;
 x=x0;
 for(x=x0;x<xp;x=x+h){
  m1=f(x,y);
  printf("\nm1=%f",m1);
  m2=f(x+h/2,y+(h*m1)/2);
  printf("\nm2=%f",m2);
  m3=f(x+h/2,y+(h*m2)/2);
  printf("\nm3=%f",m3);
  m4=f(x+h,y+h*m3);
  printf("\nm4=%f",m4);
  y=y+h/6*(m1+2*m2+2*m3+m4);
  printf("\ny=%f",y);
 }
 
 printf("Function value at x=%f is %f\n",xp,y);
 getch();
 return 0; 
 
}