Numerical solution using runge kutta with programming in c++

0 %
100 %
Information about Numerical solution using runge kutta with programming in c++

Published on March 4, 2014

Author: vjchere



Numerical Solution using Runge Kutta Method by the help of Programming in C++

Vijay Choudhary Department of Civil Engineering Poornima University, Jaipur

        Introduction Formulation of the problem Solution of the problem Approximations, errors Graphical representation Conclusion References Further Scope

A physical problem of finding how much concentration of the pollutant would be there in a lake after certain time. To find the concentration of the bacteria (pollutant), the problem is modeled as an ordinary differential equation. Runge Kutta 4th order method is used. Solutions obtained are compared with exact solutions and are graphically discussed and analyzed.

A polluted lake has an initial concentration of a bacteria of 107 parts/m3 , while the acceptable level is only 5*106 parts/m3 . The concentration of the bacteria will reduce as fresh water enters the lake. Find the concentration of the pollutant after 7 weeks. dC + 0.06C = 0 dt

The differential equation that governs the concentration C of the pollutant as a function of time (in weeks) is given by dC + 0.06 C = 0 ,C ( 0 ) =10 6 dt We Use the Runge-Kutta 4th order method and take a step size of 3.5 weeks.

#include<conio.h> #include<iostream.h> void main() { float c[10],f,t[10],h,n; cout<<"enter the initial values of concentration of bacteria "; int i; cin>>c[0]; cout<<"enter the initial value of time ";cin>>t[0]; cout<<"enter the value of time in weeks at which we want to see the concentration ";cin>>f; cout<<"enter the difference ";cin>>h; n= (f-c[0])/h;

for(i=1;i<=n;i++) { a=0 k1=-(.06*c[a]); k2=-h*(.06*(c[a]+k1/2)); k3=-h*(.06*(c[a]+k2/2)); k4=-h*(.06*(c[a]+k3)); k=(k1+2*k2+2*k3+k4)/6; y[i]=y[a]+1; a++; } Cout<<“n Table"; for(i=0;i<n;i++) { cout<<"t x=%fty=%f",val[i][0],val[i][1]); cout<<"n"); getch(); }



Figure 1 compare the exact solution with the numerical solution using Runge kutte 4th order method using different step size. It is observed that there is significant error when the calculation is done using Runge Kutte 4th order method with step size 7. This error can be minimized if we reduce the step size from 7 to 3.5. Now the numerical solution is close to the exact solution. Further reduction of step size from 3.5 to 1.75 does not bring any major error reduction. In Figure 2, we are comparing the exact results with Runge Kutte 1st order method (Euler), Runge Kutte 2nd order method( Heun) and the Runge Kutte 4th order method. It is observed that 4th order method give close approximation to exact solution than Heun’s method and Euler’s method

Book Numerical Solutions using Programming in C++, Volume I, Mittal Publication Chapter/Papers AP Azai , ., Graph Behaviour, Sindh College Of Engineering, Sindh (*paper) 64 Pages Internet James Amtoel , Contouring and Its Applications[online] document [21/02/2014]. [Accessed 4 Feb 2014].

   By the help of C++ program, it would be easy to analysis results on different cases like step size, initial conditions, boundary values, type of method etc. Like application of 1st order differential equations in Radioactive science, we can use 2nd order differential equation in practical applications. We can mould problems of Density, Population, Traffics etc into 2nd order differential equations and can study the behavior of result via graph on different inputs.

  Program 1 Program 2

Add a comment

Related presentations

Related pages

RUNGE-KUTTA 4th ORDER METHOD - C Programming Examples and ...

... Example for RUNGE-KUTTA 4th ORDER METHOD in C Programming. ... » C Programming » Numerical ... value of a given function using Runge-Kutta ...
Read more

C Program for Runge Kutta Method | Code with C

C Program for Runge Kutta Method. ... In order to solve or get numerical solution of such ordinary differential equations, ... C and C++; C# Programming;
Read more

Tutorial: Solve Runge-Kutta using C++ Program. - YouTube

Runge Kutta 4th order method for ODE2 - Duration: ... C++ Programming - Duration: ... Numerical Integration ...
Read more

16.1 Runge-Kutta Method - WebHome < TNTgroup < TWiki

Sample page from NUMERICAL RECIPES IN C: ... then using the midpoint derivative ... fourth-order Runge-Kutta method to advance the solution over an ...
Read more

Numerical Programming 2 CSE - Lehrstuhl Numerische ...

Numerical Programming 2 CSE Tutorial 2: ODE - Runge-Kutta and Multistep Methods 1) ... Combining the homogeneous solutions yields x(t) = c
Read more

Runge–Kutta methods - Wikipedia, the free encyclopedia

... is called the Runge–Kutta matrix, while the b i and c i are known ... The numerical solution to the linear ... ODE systems using implicit Runge ...
Read more

Runge-Kutta method - Rosetta Code

... NB. nextY a Calculate Yn+1 of a function using Runge-Kutta method NB. y ... runge_kutta(f); ... {t, 0, 10}] (* Numerical solution II (RK4) *) f[{t ...
Read more

Numerical Methods--Runge-Kutta Method - Delta College ...

In the last lab you learned to use Heuns's Method to generate a numerical solution to an ... the Runge-Kutta ... as the those using the ...
Read more

Differential equations - Runge-Kutta method

Runge-Kutta Method : ... The solution of this system is a 21 = c 2, W 2 = 1/ ... Solution: Example 3: Using RK method of order four find y ...
Read more