# Questions tagged [runge-kutta-methods]

For questions about the family of Runge–Kutta methods and their application in numerical methods.

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### Convert the pendulum differential equations of a second order into a first order system [closed]

I have this system of two differential equations of a second order. I got them from the Euler-Lagrange equations of double pendulum. I need to solve this using the Runge-Kutta numerical method, but my ...
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### Theory behind the Runge-Kutta method

RK4 is: \begin{aligned}k_1&=hf(x_n, y_n)\\ k_2&=hf(x_n+h/2. y_n+k_1/2)\\ k_3&=hf(x_n+h/2, y_n+k_2/2)\\ k_4&=hf(x_n+h, y_n+k_3)\\ y_{n+1}&=y_n+(k_1+2k_2+2k_3+k_4)/6\end{aligned} ...
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### Is there a generalized formula for solving systems of ODEs using 4th order Runge-Kutta?

As far as I know, we numerically solve any system by reducing it to ODEs and somehow we manage to wire the new system into the RK algorithm. I have seen solutions where there are formulas to use the ...
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### Applying a Short Force Impulse on my Object During Runge-Kutta Integration

I'm fairly familiar with Runge-Kutta ODE solvers, but recently I have begun to try to add collision to my soft-body and hard-body objects. I believe the safest and most general way to model this would ...
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### Is there a formula for the Butcher tableau for the implicit Runge Kutta method for an arbitrary number of stages?

Usually the Runge Kutta method uses 4 stages. What if I want to use an arbitrary number of stages like 300 stages, which is possible in some domains (e.g. with physics informed neural networks). Is ...
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### Solving 2nd order Diff Eqn by RK4 method

I am trying to solve 2nd order differential equation for a harmonic oscillator, $$y''+2 \beta y'+ \omega^2 y =0,$$ using RK4 method in Fortran for different values of beta n omega, program is ...
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### Study of stable oscillations of an inverted pendulum

A rigid rod of length L, supported at one end by a frictionless hinge, connected to an electric motor, through which rapid vibrations are forced hinge in the vertical direction: $$S = A\sin(\omega t)$$...
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### Runge-Kutta method for state space with input

I need to implement the 4th order Runge-Kutta method to propagate an affine non linear system. My doubt is in regards on how the input should be treated when calculating the k parameters. To be more ...
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### Question over Nodes of Runge-Kutta methods

I have (hopefully) an easy question. Suppose I have the following system (derived from an integration) $I'(t)= f(t), \, t \in [0,\pi]\\ I(0)=0$ My goal is to find $I(\pi)= \int_{0}^{\pi}f(t) \text{d}t$...
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### Pollution Problem in a room: find the time when the concentration of carbon monoxide in the room reaches $0.01\%$

A human cabin in a spaceship contains $4500\,m^3$ of air, initially free of carbon monoxide. Starting at time $t = 0$, smoke from one of the machinery room canisters containing $4\%$ carbon monoxide ...
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### NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS. [closed]

Here is a problem, this is in my compulsory list of homework, I'm trying to solve but it seems very hard for me, I've tried Taylor expansion, considered some cases of the value c,... Can you give me ...
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### Areas and adaptive-step Runge-Kutta methods

I have been using standard, non-adaptive symplectic integrators in my work for many years. This kind of integrator requires the user to provide a time-step $\Delta t$, which is kept fixed during ...
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### Periodic boundary condition and Runge-Kutta method of order 4 used in solving Partial Differential Equation(1D Wave packet equation).

I have to code for the problem of solving 1D wave packet propagation with equation given as: $$\frac{\partial u}{\partial t} + c\frac{\partial u}{\partial x} = 0$$ The initial solution is given as: u(...
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### Solving Two Body Problem With Second Order Runge-Kutta

I read a paper which want to solve two body problem with Second Order Runge-Kutta (the paper want to find optimum weight of Runge-Kutta with ANN). Here it is the two body problem. \begin{align*} x"=\...
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### runge kutta 2 in python

I am trying to solve an equation in fluid mechanics using the runge-kutta 2 method, usually it seems quite doable but in this case its with x y and z and i cant seem to make the code. Here is what i ...
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