Can Runga-Kutta method be used to solve non-linear differential equations?

Question

Consider two-body central force problem in polar co-ordinates $r,θ$.

Corresponding 2nd order differential equation is obtained by using conservation of angular momentum. This equation is :

$ d^2r/dt^2=l^2/(m^2r^3)−GM/r^2 $

$r(t)$ is the radial position of particle (of mass $m$) as a function of time $t$. $l$ is angular momentum which is constant. $G$ is gravitational constant and $M$ is mass of the heavier body, assumed to be at rest at the origin of co-ordinate system i.e. at $(r,θ)=(0,0)$

I want to solve above non-linear differential equation; it is non-linear since dependent variable $r$ has powers -3 and -2 on RHS.

Can I use 4-order Runge-Kutta method to solve this equation ?

Extra Note: Actually we have two different 2-order differential equations (coupled) : one for $r$ and another for $θ$. Conservation of angular momentum de-couples them and reduces to one equation given above. Also if we try to solve the above 1-Dim equation analytically, we end up with a solution of the form $t(r)$ i.e. time is function of $r$. So we have to invert that into $r(t)$. This inversion process can be extremely difficult in practice. Please see standard textbook on classical mechanics e.g. by Goldstein (Chapter 3).

Here Initial conditions should be on $r$ and $dr/dt$, if we want to solve numerically. But if we want to solve analytically we need initial conditions as Total Energy and Angular momentum of the mass $m$. I am confused here: do I need use all initial conditions i.e. energy, angular momentum, $r(t=0)$ and $dr/dt$ at $t=0$ ?

Answer 1

Runga-Kutta schemes are multistage recipes for numerical discretizations of temporal derivatives. That is they tell to how to solve equations in the form $\dot{\mathbf{y}} = \mathbf{f}(\mathbf(y,t)$. In order to achieve the advertised accuracy the right hand side should be a sufficiently smooth function of $y$, and nonlinearity can complicate the question of stability for explicit methods like RK4, but otherwise I see no reason RK4 can't be used with a sufficiently small tilmestep.

That said, as you've described, this is a problem with a Hamiltonian formulation, so there are more appropriate symplectic methods like Stormer-Verlet that are worth looking in to.

Regarding the question of initial conditions, you should be able to calculate the energy in the system given $r(0)$ and $\dot{r}(0)$, so specifying all three would lead to an over constrained system (i.e. you only need two, but are free to pick any pair).

Answer 2

Yes. Runge-Kutta techniques like the Dormand-Prince 4/5 method (ode45) are the standard numerical techniques for solving nonlinear differential equations. Of course there are many more methods available, but all of them are designed to be used on nonlinear equations where no analytical solution exists.