How do I specify derivatives in C++ for use in the 4th order Runge-Kutta method?

[brokenlynx]

I'm just trying to understand how exactly derivatives are to be specified in C++ for use in the 4th order Runge-Kutta method.

I am implementing the method in C++ (though this part of my program isn't complete yet) however what I am looking to do is first specify the derivatives of the variables in the two first-order differential equations I am working with:

θ' = ω
ω' = − g⁄R sin θ

(for modelling the motion of a single pendulum)

double theta - initial angular displacement (Radians)
double R - length of Pendulum Rod (Metres)
double g - gravitational constant (Distance/Time^2)

I just need a little help understanding how to specify the derivatives and basically what I'm doing when I am. (I'm familiar with calculus just not numerical solving algorithms).

Thanx.

 

[EnumaElish]

I will assume that the solution y = (y1, y2) = (θ, ω) you are seeking is defined in a "t x ANGLE²" space where θ $\in$ ANGLE and ω $\in$ ANGLE and variable t (e.g., time) is implicit -- that is, t does not appear as an argument of θ or ω. If that is not right, please post so.

Given the initial value y(0) = (y1(0), y2(0)) = (θ(0), ω(0)), and an (arbitrarily small) step size h, the 4 functional steps are:

k1(0) = (k11(0), k12(0)) = (θ'(0), ω'(0)) = (ω(0), −g⁄R sin θ(0));

k2(0) = (k21(0), k22(0)) = (θ'(0)+h k11(0)/2, ω'(0)+h k12(0)/2) = (ω(0)+hω(0)/2, −g⁄R sin θ(0)+h[−g⁄R sin θ(0)]/2);

Then define k3(0) = (k31(0), k32(0)) in terms of k2(0) = (k21(0), k22(0)) as described here: http://en.wikipedia.org/wiki/Runge-kutta#The_classical_fourth-order_Runge.E2.80.93Kutta_method;

Then define k4(0) = (k41(0), k42(0)) in terms of k3(0) = (k31(0), k32(0)) as described here: http://en.wikipedia.org/wiki/Runge-kutta#The_classical_fourth-order_Runge.E2.80.93Kutta_method.

Finally, calculate the weighted average y(1) = (θ(1), ω(1)) = y(0) + [k1(0) + 2k2(0) + 2k3(0) + k4(0)]h/6. And you are on your way to calculate y(2), ..., y(N).

 

[tacman]

In order to use the 4th order Runge-Kutta method to solve a system of differential equations, you need to first specify the derivatives of the variables in the equations. In your case, the two first-order differential equations you are working with are:

θ' = ω
ω' = − g⁄R sin θ

To specify these derivatives in C++, you will need to define two functions: one for θ' and one for ω'. These functions will take in the current values of θ and ω, and return the values of θ' and ω' respectively. The code would look something like this:

// Function for θ'
double theta_prime(double theta, double omega, double R, double g) {
return omega;
}

// Function for ω'
double omega_prime(double theta, double omega, double R, double g) {
return (-g/R) * sin(theta);
}

Once you have these functions defined, you can use them in your implementation of the 4th order Runge-Kutta method. This method involves using a series of calculations to approximate the values of θ and ω at each time step. You can refer to a numerical methods textbook or online resources for the exact implementation of the algorithm in C++.

In summary, to specify derivatives in C++ for use in the 4th order Runge-Kutta method, you will need to define functions for each derivative in your system of differential equations, and use those functions in your implementation of the method. I hope this helps clarify your understanding of how derivatives are specified in this context.