Questions tagged [runge-kutta-methods]
For questions about the family of Runge–Kutta methods and their application in numerical methods.
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Why RK3 is more stable than forward euler
I'm working on hyperbolic equations.
I implemented RK1 (Euler), RK2 and RK3 for the convection equation with a central scheme in space. For a smooth solution, I have a perfect solution while for RK1 ...
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3
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Numerically solve a system of two equations using fourth-order Runge-Kutta
I intend to solve the following system
$$
\left\{
\begin{array}{l}
\frac{du}{dt} = \frac{- \cos(v) \cos(u)bc+\sin(v)a}{cab} :=f(u, v, t) \\
\frac{dv}{dt}=\frac{(\sin(v) \cos(u)bc+ \cos(v)a) \cot(u)}{...
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Reference for Shooting Method
Consider the following setup. We have a second order boundary value problem:
$$\dfrac{d^2y}{dx^2}=F(x,y,dy/dx);\qquad y(x_0)=y_0,\quad y(x_f)=y_f.$$
A numerical approach is to almost first write as ...
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Error of the Runge-Kutta method
In my university excercise I've run into an ODE system
$$\begin{cases}
\frac{d\rho}{dr} = - \frac{m}{r^2}, \\
\frac{dm}{dr} = \rho r^2, \\
\rho(0)=1, m(0) = 0.
\end{cases}$$
The analytical solution to ...
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Consistency of Runge-Kutta methods
Consider the Runge-Kutta method given by
\begin{equation*}
y_{n+1} = y_n + \Delta t \phi(t_n,y_n,\Delta t),
\end{equation*}
with
\begin{equation*}
\phi(t_n, y_n, \Delta t) = \sum_{i=1}^s b_i ...
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How to solve this difficult differential equation with Runge-Kutta methods?
Given the following differential equation (it can be seen as a matrix differential equation):
$$\begin{cases}
x'=\dfrac{dx}{d\phi}=\dfrac{ \cos(\phi) }{ 2+\beta z-(1/x) \sin(\phi) } \\
z'=\dfrac{dz}{d\...
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Solving $y'=(x-1)y$ $y(0)+2$ using $RK2$
Solve the following cauchy problem: $y'=(x-1)y$ and also given $y(0)=2$
using the runge kutta 2 method $(RK2)$ for $\alpha=0.2$ on the interval $[0,0.6]$ with step $h=0.2$
The answer in the book:
for ...
- 913
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Computing the relative error of two Runge Kutta Methods for Convergence Analysis
I am currently endeavoring to assess the relative error between the classical Runge-Kutta (RK4) method and another RK variant. I've opted to employ the Ordinary Differential Equation (ODE) governing a ...
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3N low storage method for ssprk(5,4)
I am trying to understand how the SSPRK(5,4) integrator can be implemented using 3N storage registers (as is done in Athena++). I cannot seem to find a reference explicitly stating the algoirthm ...
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Simulation of a pendulum on a spinning Disk
I don't know if this is the right place to ask this Question, but I have previously asked a similar question where i asked how to write a simulation on this phenomenon. I got a great answer with a ...
- 119
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Runge Kutta with multiple variables
How can I apply 4th order Runge Kutta to a function that requires multiple inputs (instead of $\dot x = f(x, t)$, something like $\dot x = f(a, b, c, d, t)$). For $\dot x = f(x, t)$, the Runge Kutta ...
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What is a converging explicit Nyström method for an object experiencing friction?
Consider the dynamic simulation of an object that is sliding across a level surface and experiencing friction. The friction is a lower kinetic friction if the object is sliding faster than some ...
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Standard way to do Runge-Kutta (4th order) for coupled ODE's in Python?
I am somewhat familiar with using RK4 for coupled ODE's, I found a very elegant way (in my opinion) to utilize it in Python, like so:
...
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How to find out the Runge-Kutta 4 constants for numerically evaluating an nth order IVP?
I know how to use the RK 4 method for a first order differential equation of the form:
$$y' = f(x, y(x))$$
$$y_{k+1} = y_k + (G_1 + 2G_2 + 2G_3 + G_4)*dx/6$$
where $G_1 + 2G_2 + 2G_3 + G_4$ are ...
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How do I obtain the standard ODE for this system of equations?
I am attempting to numerically simulate a specific physical model, and I have obtained the system of equations using the Lagrange method. I'm not sure if this question is better suited for the Physics ...
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