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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 ...
Taendyr's user avatar
1 vote
3 answers
68 views

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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5 votes
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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 ...
Somestudent01's user avatar
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1 answer
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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\...
fede1602's user avatar
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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 ...
Adamrk's user avatar
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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 ...
Can's user avatar
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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 ...
BrayA's user avatar
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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 ...
Mo711's user avatar
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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 ...
MaximeJaccon's user avatar
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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 ...
Sibbo's user avatar
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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: ...
Vox Winters's user avatar
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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 ...
Ajaykrishnan R's user avatar
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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 ...
Gum's user avatar
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Predictor Corrector Scheme for Implicit Runge Kutta

I want to solve an ODE system : $$ \frac{dy}{dt} = f(y, t) $$ Since my application requires method to be symplectic, I am using an implicit runge kutta method. $$ y_{n+1} = y_n + h\sum_{i=1}^s{b_iK_i} ...
Chandan Gupta's user avatar
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Showing that an explicit s-stage RK method with its order of accuracy higher than s

I was asked to show an s-stage explicit Runge-Kutta Method cannot obtain accuracy higher than $s$. It suffices to consider autonomous system $y' = f(y)$ (otherwise by introducing a new variable $t$, ...
Stack_Underflow's user avatar
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2 answers
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Question about what the brackets on the superscript on this question means?

Good day :) I would just like to ask about what the brackets on the superscript on this question means? Thank you for your help.
user123456098's user avatar
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How do I expand the fourth-order Runge-Kutta method to the seventh-order?

I'm being asked this question and I don't quite understand how to do it. I kinda know how the fourth-order Runge-Kutta method works. $K_1 = hf(x_n,y_n)$ $K_2 = hf(x_n+\frac{h}{2},y_n+\frac{k_1}{2})$ $...
mEXsACHINE's user avatar
1 vote
1 answer
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How to numerically solve an ODE with a critical point

I'm trying to use fourth-order runge kutta method to solve the equation: $$\frac{y-1}{2y}\frac{dy}{dx}=(3+\frac{y}{2})\cdot\frac{1-\frac{1}{x}}{\frac{3}{2}x+2}$$ boundary conditions: $$y\to\infty\ as\ ...
Ian's user avatar
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2 answers
225 views

Showing Runge-Kutta implicit method local truncation error

Consider the implicit Runge-Kutta method: \begin{equation*} y_{n+1} = y_n + hf\left(t_n + \frac{2}{3}h, \frac{1}{3}(y_n + 2y_{n+1})\right) \end{equation*} a) Show that the local truncation error ...
Tomas Escobar Rivera's user avatar
1 vote
1 answer
67 views

Runge Kutta 4th order with coupled equations where derivatives are linked [closed]

I have 2 coupled second order equations as below: $$ \ddot{y}(t) + a\dot{y}(t) + by(t)=q(t) $$ $$ \ddot{q}(t) + c\dot{q}(t) + dq(t) = A\ddot{y}(t)$$ I'm wondering if it is possible to solve this ...
Avatar36's user avatar
2 votes
1 answer
128 views

Total energy oscillations with Runge-Kutta 4th order method - how to avoid them?

Consider a free ideal pendulum, which obeys the equations: $$\frac{d\varphi}{dt}=p \\ \frac{dp}{dt}=-\omega_{0}^{2}\sin\varphi$$ I am applying two 4th order Runge-Kutta schemes: the usual explicit one,...
Yuriy S's user avatar
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Why the weights are 1-2-2-1 in Runge-Kutta method? [closed]

The well-known 4th order Runge-Kutta formula for the ODE $y'=f(x,y(x)$ is given by $$y_{n+1} = y_{n} + \dfrac{1}{6}h(K_1 + 2 K_2 + 2K_3 +K_4),$$ where $K_1,K_2,K_3,K_4$ are essentially approximations ...
Hanh's user avatar
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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 $L$...
tommy1996q's user avatar
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0 votes
2 answers
90 views

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 ...
Octavius's user avatar
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0 votes
1 answer
68 views

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} \...
Henry T.'s user avatar
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0 answers
50 views

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-\...
user1942348's user avatar
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0 answers
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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.
Vick's user avatar
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2 votes
1 answer
197 views

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 ...
user122424's user avatar
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-2 votes
1 answer
99 views

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 $...
user122424's user avatar
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0 votes
0 answers
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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 ...
Marco's user avatar
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0 answers
73 views

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 ...
user122424's user avatar
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0 votes
1 answer
43 views

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 ...
Ian Holmes's user avatar
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46 views

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 ...
user avatar
2 votes
0 answers
126 views

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-...
Amatics's user avatar
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3 votes
0 answers
128 views

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, ...
TriJB's user avatar
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3 votes
1 answer
612 views

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-...
Amatics's user avatar
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0 votes
1 answer
107 views

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 ...
Amatics's user avatar
  • 55
0 votes
0 answers
67 views

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 ...
Oliver Noxtraxs's user avatar
0 votes
1 answer
115 views

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 $(...
user122424's user avatar
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0 votes
0 answers
44 views

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 ...
Makogan's user avatar
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0 votes
0 answers
107 views

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 $y_{p_x,p_y}(x)$. ...
Debamalya Dutta's user avatar
0 votes
2 answers
108 views

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 ...
trex's user avatar
  • 85
2 votes
1 answer
52 views

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 ...
WnGatRC456's user avatar
-4 votes
1 answer
127 views

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{...
Furkan Bostancı's user avatar
2 votes
3 answers
107 views

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. ...
jageren7's user avatar
0 votes
1 answer
358 views

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 ...
Ali Kıral's user avatar
1 vote
0 answers
156 views

Why is my fourth order Runge Kutta method not accurate

...
Eric L.'s user avatar
  • 119
0 votes
1 answer
88 views

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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