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Questions tagged [runge-kutta-methods]

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

Topics in this tag include:

  • The standard 4-stage Runge–Kutta method.
  • Other iterative numerical methods that belong to the family of Runge–Kutta methods.
  • The application of Runge–Kutta method in providing numerical solutions to mathematical models.

When posting in this tag, consider attaching these more general tags below, if they are applicable:

476 questions
39
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2 answers

Explanation and proof of the 4th order Runge-Kutta method

The 4th order Runge-Kutta (RK4) method is a numerical technique used to solve ordinary differential equations (ODEs) of the following form $$\frac{dy}{dx} = f(x,y), \qquad y(0)=y_0$$ It gives $y_{i+1}$ in the form $$y_{i+1} =…
gen
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38
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5 answers

Help with using the Runge-Kutta 4th order method on a system of 2 first order ODE's.

The original ODE I had was $$ \frac{d^2y}{dx^2}+\frac{dy}{dx}-6y=0$$ with $y(0)=3$ and $y'(0)=1$. Now I can solve this by hand and obtain that $y(1) = 14.82789927$. However I wish to use the 4th order Runge-Kutta method, so I have the…
Michael
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1 answer

Runge-Kutta method for PDE

I consider certain partial differential equation (PDE). For example, let it be heat equation $$u_t = u_{xx}$$ I want to apply numerical Runge-Kutta method for solving it. As a first step I approximate $u_{xx}$ with difference scheme of several…
newt
  • 391
9
votes
2 answers

What's the difference between explicit and implicit Runge-Kutta methods?

I have been working on numerical analysis, just as a hobby. I am only aware of the basic fourth order Runge-Kutta method in order to solve problems. When I was digging deep into it, I found there are types, mainly explicit and implicit, but I was…
Rahul
  • 263
8
votes
2 answers

Runge-Kutta method and step doubling

I am studying Runge-Kutta and step size control and came up with a few doubts. Because they are related with this integration method, I will divide it in two parts. First, allow me to introduce the problem. $1^{st}$ part - questions about…
Sos
  • 343
8
votes
1 answer

What's the motivation for Runge-Kutta methods?

Recently, I have been taking a course on ODEs and learning Runge-Kutta methods. To be specific, the 4th order Runge-Kutta method on solving initial value problems. My instructor and the textbook told me the formula but didn't say anything about the…
Hgtcl
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8
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2 answers

Differences between Quaternion integration methods

I've implemented a Quaternion Kalman filter and i have the choice between multiple way to integrate angular velocities. The goal is to predict futur orientation $q^{n+1}$ from current orientation $q^{n}$ and angular velocity $\vec{r}$. During the…
jcolafrancesco
  • 153
7
votes
1 answer

difference between runge kutta methods of same order

I recently read about Runge-Kutta methods for solving differential equations. So far I understood the idea but up to now nobody could answer the following question: If we consider the explicit RK methods of order 4, we get for example the classical…
flawr
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6
votes
2 answers

How to verify the order of DOPRI Runge-Kutta method?

I've written code in Fortran based on the RK8(7)-13 method by Dormand and Prince to solve the system $\mathbf{y}'=\mathbf{f}(t,\mathbf{y})$. The method is Runge-Kutta 8$^\text{th}$ order with an embedded 7$^\text{th}$ order method; I want to verify…
Mr. G
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6
votes
2 answers

Prove that Runge Kutta Method (RK4) is of Order 4

Please somebody help me, recently we have been studying numerical methods for solving ODEs and we went over proofs for the Euler method being order 1 and Huen’s method being order 2. But our lecturer didn’t cover why the Runge Kutta method (RK4) is…
Dani_5040
  • 117
6
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2 answers

Error in classical RK method. The result went to infinity.

I used classical ($4^\text{th}$ order) Runge-Kutta method to solve the ODE $$y'=5e^{5t}(y-t)^2+1,0\leq t\leq 1, \ y(0)=-1.$$ $h$ is the step size. When $h=0.2$, I got a good approximation of the solution. However, when $h=0.25$, the result went to…
learner
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5
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1 answer

Runge-Kutta method accuracy

I got a Runge-Kutta method here and I solve this system using it. So here's Runge-Kutta stuff \begin{align} k_1 &= f(t_n, y_n) \\ k_2 &= f(t_n + h/2, y_n + hk_1/2) \\ k_3 &= f(t_n+h, y_n - hk_1 + 2hk_2) \\\hline y_{n+1} &= y_n + h(k_1 + 4k_2…
Danil Gholtsman
  • 243
5
votes
0 answers

Reference for Shooting Method

Consider the following setup. We have a second order boundary value problem: $$\dfrac{d^2y}{dx^2}=F(x,y,dy/dx);\qquad y(x_0)=y_0,\quad y(x_f)=y_f.$$ A numerical approach is to almost first write as two first order…
JP McCarthy
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5
votes
1 answer

How to choose the order of a Runge-Kutta method?

I have seen that Runge-Kutta's methods are a family of methods used to approximate the solution of an initial value problem. I have also seen that they are classified depending on their order (with the second-order R-K being the Euler's Modified…
Quaerendo
  • 606
5
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0 answers

error sensitivity analysis of Runge - Kutta method

In Runge - Kutta - Fehlberg methods, sometimes and in some cases the answer depends on the method we define the error and also on the magnitude of the error. In the case I am working on, there are several "zero crossings"; which is problematic…
Radmarz Hosseinie
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