The Runge–Kutta methods are a set of numerical methods for ordinary differential equations for the approximation of their solutions.
Questions tagged [runge-kutta]
162 questions
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Runge-Kutta methods, higher derivative methods, and collocation methods
Consider an ODE system
$$\dot x = f(t, x), \quad x(0) = \xi.$$
A collocation method to solve this ODE (1) assumes that $x$ can be approximated as a polynomial
$x(t) \approx \sum_kx_kp_k(t)$ and (2) chooses the coefficients $\{x_k\}$ so that the…
Daniel Shapero
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How to estimate the local error and the global error for Runge-Kutta method
How to estimate the local error and the global error for Runge-Kutta method used for solve a system of differential equations in practice?
I use Richardson extrapolation for select a adaptive step [Solving Ordinary Differential Equations I -…
Queue Overflow
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Runge Kutta Procedures for Incompressible Navier Stokes
I was playing a little bit with the Runge-Kutta procedure for the Incompressible Navier-Stokes equation and came up with something strange, so I would like to know where I'm wrong or doing something I shouldn't.
Let's consider
$u_i = u^n + \Delta t…
Marco
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What is the derivation of the values of a1, a2, p1 and 11 in the Second Order Runge Kutta Method?
So currently I am studying about the Runge Kutta Second Order Method used to estimate first order ordinary differential equations. The following show the formulas.
$$ y_{i+1} = y_i + (a_1k_1+a_2k_2)h $$
$$ k_1 = f(x_i,y_i) $$
$$ k_2 = f(x_i + p,y_i…
AndroidV11
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4th order Runge-Kutta Method for Driven Damped Pendulum
Although I've been looking everywhere, I have been unable to find an answer to my question so here it is. For a driven damped pendulum the equation of motion in dimensionless units is,
$$\alpha(\omega,\theta,t)=-c\ \omega -\sin \theta +F(t).$$
My…
Josh
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Shooting Method with RK4 and Thermal Radiation
I am attempting to numerically solve the following problem. I decompose it into a system of two first order ODEs and then solve via the shooting method. I use the fourth order Runge-Kutta (RK4) method to solve each iteration of the shooting…
Nukesub
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Why is it assumed that $c_i = \sum_{j=1}^sa_{i,j}$ in the butcher tableau of a RK-method?
In my textbook it is stated that we make a "simplifying assumption"
$$c_i = \sum_{j=1}^sa_{i,j}, $$
where $c_i, a_{i,j}$ are the constants in the butcher tableau.
What's the relevancy of this assumption? Is it only for explicit RK-methods?
Heuristics
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Runge-Kutta when time dependence not known
As a simple exercise for a class in computational methods for physics, we've learned how to implement the RK4 (Runge-Kutta 4th-order) algorithm for a very simple exponential decay. E.g. the function y(t) is governed by
Based on the RK4 definition…
WindowsNT
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How and when to use IMEX runge-kutta
i have come across method of IMEX Runge-Kutta written by Kennedy and Carpenter in july 2001. The method consist of two kind that is "implicit and explicit"
can i use the ERK@explicit part only to solve the intial value problem, as if i treat the…
MamaKyle
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