Questions tagged [runge-kutta-methods]
For questions about the family of Runge–Kutta methods and their application in numerical methods.
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Global truncation error of backward Euler method
It's often found in books that the global truncation error of the forward Euler method applied to $\dot{y} (t) = f(t, y(t))$ is given by something like
$$ \frac{\exp(LT) -1}{L} \frac{Mh}{2},$$
with $L$...
- 3,170
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2
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Runge Kutta Method for in $1+1$ dimension
Given a partial differential equation
$\partial_t u(t,x) = F(t,x,u,u')$
Suppose I know the functions $u(t_0,x)$ and $u'(t_0,x)$ at some point $t_0$ for all $x$. In order to obtain the function $u$ at ...
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Order of a method to solve an ODE
In my exercise the goal is to find a numerical method to solve:
$$
\dot{q}=p, \quad \dot{p}=-\omega^2q
$$
the final method (if I did everything correctly) is:
$$
\begin{pmatrix} q_{k+1} \\ p_{k+1} \...
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Non-standard finite difference for a reaction diffusion system
I want to discretize the following reaction diffusion system:
$\frac{\partial u(x,y,t)}{dt}=\nabla ^2u+ u(1-u)-\frac{uv}{u+\alpha v}$,
$\frac{\partial v(x,y,t)}{dt}=d\nabla ^2v+ \delta v\left(\beta-\...
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How to apply Runge Kutta method to this equation
I'm trying to solve three dimensional heat equation. I use the method of lines to discretize the spatial part,and choose five point stencil. I got the following equation:
$$ \frac{\mathrm{d} U}{\...
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How is the E coefficients calculated or generated for the Radau IIA 5th Order?
With regard to Scipy, the E coefficients are written out. I've seen this from different sources as well. However, I cannot find any reference to how the E coefficients were computed.
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Explicit Runge-Kutta method for solving ODE
I would like to understand in some detail why this Runge-Kutta method is explicit if and only if $c_1=0$.
Why in this case the r.h.s. doesn't contain $y_{n+1}$ ?
EDIT $c_1=0$
EDIT 2
I would also like ...
- 3,810
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1
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The Taylors expansions of Butcher's B-series (for Runge-Kutta order conditions) [closed]
In the Butchher's book numerical methods for ODE I do not understand in the proof how do we obtain the coefficient in the formula $311d$ at $F(t)(y(x_0))h^{|t|}$ from $$\frac{1}{t!}\xi^{|t|}$$ where $...
- 3,810
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Navier Stokes runge kutta question
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 ...
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Stages vs. Order in numerical analysis for ODE
I'm reading this interesting book by Butcher.
I do not understand what are the definitions of stages and orders of numerical methods.
I think that these 2 terms are somewhat overloaded.
Which page for ...
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Systematic approximation to derivative of Runge-Kutta integral with respect to a parameter?
Suppose I have a differential equation
$\frac{d}{dt}y(t,\lambda) = f(y(t,\lambda),\lambda)$
where $\lambda$ is some parameter of the system.
The initial value $y(0,\lambda)$ is specified.
I am looking ...
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What is this modified Runge-Kutta method and what error estimates exist?
In RK methods you are given a (small) time increment $h>0$, then you compute an approximate solution $(y_n)_n$ of an (autonomous, say) ODE $y'=X(y)$ by computing a well-chosen linear combination of ...
user142290
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Backward Euler Method - solving algebraically
One of my notes mention that it is "not allowed to solve a Backward Euler Method algebraically"
I have my doubts; for example
$y'(t) = y(t) + 2$
$y(0) = 1$
Then we can compute, with $h = 0.1$...
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Runge-Kutta-4 Solution for a highly NON LINEAR system of ODE
I recently have been working on models for underwater bubble dynamics which are expressed by highly Non-linear system of ODE's. I have been trying to formulate a numerical method scheme using Runge-...
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Difference between iterative methods (Gauss-Seidel, Newton-like methods) and Runge-Kutta methods for ODE / PDE solutions
This is a general question to try and understand conceptually (in laymans (i.e. engineers) terms) the differences between the above solution methods for things like solving ODEs and PDEs. For context, ...
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How do I input a vector form of ODE's on Runge-Kutta-4 for generality to solve in matlab
I have come across a system of ODE's that are written on vector/Matrix format such that; $Ax'=b$
For simplicity, say the system of ODE's has a vector $x'$ containing first order derivatives of 2-...
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How to input First Order Non Linear ODE that are dependent on each other into Runge-Kutta [closed]
I am working on ODE's that are highly non linear (even at only their first order expressions). The governing equations of motion describes a complex phenomenon of bubbles underwater. Runge Kutta 4 ...
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A-stability of implicit Runge Kutta methods
Are implicit Runge Kutta methods always $A$-stable? I know that explicit and implicit Runge Kutta methods are always $0$-stable and explicit Runge Kutta methods are never $A$-stable. But do implicit ...
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Existence and uniqueness theorem applying the Runge-Kutta formula
I want prove that the problem
$$y'=f(x,y)$$
$$y(x_0)=y_0$$
It has a unique solution if $f$ is continuous in $[a,b] \times \mathbb{R}^n$ and it satisfies the Lipschitz condition with respect to the ...
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Runge-Kutta method, the tables
I have a problem in this excellent book in the derivation of the Runge-Kutta methods,
The numerical analysis of ODE.
My wish is to understand how the data as in [*] below are obtained from item $(...
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Trying to get region of stability of RADUA IIA
The method can be turned from its Butcher table into this set of explicit expressions:
$$k_1 = f\bigg(t_n + \frac{1}{3}h, y_n + \frac{5}{12}hk_1 + \frac{-1}{12}hk_2\bigg)$$
$$k_2 = f\bigg(t_n + h, y_n ...
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What numerical integration method can solve a integral over a function that depends on a differential equation without an analytic solution?
Suppose we have a differential equation of the form $\frac{dy}{dx}=f_{p_x,p_y}(y,x)$. After solving this differential equation for particular values of $p_x$ and $p_y$, we obtain $y_{p_x,p_y}(x)$. ...
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Velocity field integration
Suppose we have a velocity field $$\mathbb{v}=\begin{bmatrix}v_x(x,y)\\ v_y(x,y)\end{bmatrix}$$ and the relation
$$\frac{d\mathbb{v}}{dt}=f(\mathbb{v},t).$$ I am currently using Runge-Kutta's Method ...
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1
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Rules for Choosing Bounds and Initial Conditions when Using 2nd Order Runge Kutta Methods
I have a question regarding 2nd order Runge-Kutta methods, specifically where it regards the bounds of the solution.
Let's say I have to solve a 1st order ODE $\frac{dy}{dx}=f(x,y)$ numerically using ...
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Runge-Kutta-Fehlberg $4(5)$ method adaptive size $h$ - iterating too much
I tried to solve baryocentric two body problem with Runge-Kutta-Fehlberg with adaptive size method.
I have two differential equations
$$ \frac{d^2x}{dt^2} = - \frac{\mu}{(x^2 + y^2)^{3/2}} $$
$$ \frac{...
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How to solve this autonomous DE with RK 4?
I have this equation :
$$\frac{d\alpha}{dz} = - \frac{dr}{dz} * \frac{\tan(\alpha)}r $$
I searched for some similar examples but non of these equations was like this one.
I'm confused about this one. ...
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An adaptive step size solver for an ODE
I am trying solve an ordinary differential equation numerically, $\,dy/dt=10e^{-(t-2)^2/2(0.075)^2}-0.6y\,\,$ with an initial value and initial step size between $t=0\, and\, t=4$.
In my code I ...
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Stability function at infinity of a runge-kutta method
Recently, I have been studying some bits of numerical analysis and
have managed to derive the stability function of a Runge-Kutta method.
What I got is as follows
$R\left(z\right)=1+zb^{\top}\left(I-...
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1
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Runge-Kutta Method [ Butcher Tableau ] Show $A$-Stability.
We consider the following Butcher Tableau:
\begin{array}{c|cc}
\frac{1}{3} & \frac{5}{12}& -\frac{1}{12} \\
1 & \frac{3}{4} & \frac{1}{4}\\
\hline
& \frac{3}{4} & \frac{1}{4}...
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Variable-step Runge-Kutta methods, Fehlberg vs Dormand-Prince: why is the order reversed?
Dormand-Prince and Fehlberg are two popular Runge-Kutta embedded methods for ODE integration with adaptive stepsize.
The former one estimates the error with the lower-order method and steps forward ...
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1
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How to numerically solve a system of ODEs using 4th order Runge-Kutta method integrating backwards?
I am trying to solve a system of two 2nd order ODEs using the 4th order Runge-Kutta (RK4) method. The equations are of the form:
$$\frac{d^2r}{dt^2}=f(r,\theta,\dot{r},\dot{\theta}),$$
$$\frac{d^2\...
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Advanced Runge-Kutta vs Symplectic integrators
Symplectic integrators are build from compositions of discrete symplectomorphisms. Symplectic integrators do not conserve energy, but energy error is bounded. Since non-linear time translations are ...
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In Adaptive ODE Methods, should the more accurate method be accepted or not?
I'm a little unsure if there are general rules for which method to use in an adaptive ODE scheme that uses two methods of different accuracies to estimate the error.
For example, the approach I am ...
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How to simulate a control system with RK4
My question is simple yet I could not find answer on the internet. When I have a system described by its differential equation, I can simulate its states from time steps to time steps with RK4. But ...
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Doing a Runge-Kutta calculation for deceleration due to drag
Sorry, I'm new to Runge-Kutta calculations, but I'm working on a game and I believe they're what I need to better model deceleration due to drag. Basically, I have an engine producing engineForce, ...
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Runge-Kutta method for heat equation
I am trying to solve the following heat equation for a rod using an explicit Runge-Kutta method in time:
$$
\frac{\partial T_{i}}{\partial t}=-u \frac{\partial T_i}{\partial z}+k \frac{\partial^{2} ...
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Richardson extrapolation of 4th order Runge-Kutta
I want to find the Richardson extrapolation of the 4th order Runga-Kutta method. I found a formula on Wikipedia for a general approximation and then I tried applying it to the specific method. I got ...
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Implicit numerical ODE solvers: still unconditionally stable if Newton iterations limited?
Implicit ODE solvers like Backward Euler are often described as unconditionally stable, which I believe means that the solution never blows up regardless of how large the solver step size is chosen.
...
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Nonlinear ODE system solver (Runge-Kutta)
I`m trying to solve the following nonlinear-system(please see the pic attached) but I´m truly lost.
EQUATIONS
I need to create one method called solve_prec, which solves the system. I dont know how to ...
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Using Runge-Kutta 4 to approximate y(t)
I want to use Runge-Kutta 4 to solve the IVP
$$
y' = y, \ y(0) = 33
$$
numerically and I want to find $y(2^k)$ where $k = 1,2,3,4...$ with an accuracy of two correct decimals.
I've created a python ...
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Estimating pi with Euler's method and Runge-Kutta 4
I'm trying to estimate pi by solving the IVP
$y'' + y = 0$
where
$y(0) = 1, y'(0) = 0$
numerically by defining $\frac{\pi}{2}$ as the first value on t such that $y(t) = 0$
I'm trying to solve this ...
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How to design a gravitational slingshot
I've created a simple solar system in python containing the sun, earth and jupiter by solving a system of differential equations with Runge-Kutta 4. Now I have a space ship which is close to earth, ...
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Using RK4 for ODE's with variable coefficients: Forced vibrations.
I'm trying to solve the second order ODE of a simple damped oscillator with forced motion using RK4.
I've already solved it with MATLAB's ode45 and i have the analytical solution, so i should be able ...
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Runge Kutta Methods and Discontinuities
Given a second order ODE
\begin{equation*}
\ddot x(t) = \begin{cases}
a_1 \text{ for } [t_k, t_{k+1})
\\
a_2 \text{ for } [t_{k+1}, t_{k+2}]
\end{cases}
\end{...
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Runge-kutta fourth order for 3 coupled second order equations.
Someone, please help me by checking whether the steps of the application of RK4 in my calculation is correct or not. I could not find any paper/books/write with similar problems or examples. ...
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Logistic equation - How to go from continuous form to discrete form?
Logistic equation in continuous form:
$\frac{\mathrm{d} y}{\mathrm{d} t} = ry(1 - ay)$ (Autonomous Differential Equations and Population Dynamics, equation 6 in Boyce Diprima's book, eleventh Edition)...
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Help with Runge-Kutta Order 4th with a system of 3 equations
I have been trying to solve a system of equations with Runge-Kutta Order 4th but every time I try to run the program it says running but it doesn't load anything so I can't check if it is good or not. ...
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I don't understand how to properly use runge kutta
So I am trying to use runge kutta 4, to more acurately calculate forces. I am using code from github and I used their example of rabits and wolves populations. In IntegratorLSODE.cs, in lotkaVolterra()...
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Computing product of matrix exponential with vector using Runge Kutta
I have a matrix $A$ and a vector $b$. I need to numerically calculate $e^{t A}b$.
That can be seen as the solution of the initial value problem
$$
\frac{dy}{dt} = A y; \;\;\; y(0)=b
$$
Therefore I can ...
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