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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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19 views

On a second order non-linear differential equation

How can one solve following non-linear differential equation (in term of $z$)? $$∂^2 y/∂z^2 +2ib ∂y/∂z -d y^3 =0$$ If I knew the boundary conditions $y=a$ and $y'=0$ at $z=0$ and I knew the ...
0
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1answer
31 views

Consistency improved Euler method

I have the butcher tablaeu for the improved Euler method \begin{array} {cc|c} 0 & 0 & 0 \\ 1 &0 & 1 \\ \hline \frac{1}{2} &\frac{1}{2} \end{array} I need to show that this ...
1
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1answer
35 views

How to input this type of Nonlinear ODE into Runge-Kutta

Say I have a differential equation, such as $$y'(t)+y'(t)^3 -t^5+350=0$$ with some initial condition. How can we take this nonlinear ODE and plug it into a system that runge-kutta can deal with? I ...
0
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1answer
36 views

Two-Dimensional Runge-Kutta

I have just started getting into ODEs, and have come across the Runge-Kutta method for numerically solving them. However, in playing around with them to model hypothetical situations, I came across ...
1
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1answer
45 views

Why use Classic fourth-order Runge-Kutta over the 3/8-rule?

I have been reading about Runge-Kutta methods, particularly the "classical" fourth-order method. When it is talked about, the 3/8th rule is often mentioned. For example, in this document, the ...
0
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1answer
18 views

Show solution from Runge Kutta program is correct to 5 decimal places

I have built a program which implements RK4 method to solve ODEs. I want to show my program can find a specific value correct to 5 decimal places. How would I prove this?
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21 views

Adams Bashforth/Moulton, Numerical scheme in multi step method

There is this question that has bothered me for a while. For example, when we do numerical method in differential equation, namely the multi step method to approximate some solution of a DE, we can ...
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9 views

Verify that the recurrence formula for the general second order Runge-Kutta method is $y_{i+1}=y_i+h_i[(1-\beta)k_1+\beta k_2]$

We look forward the discretization formula $$ \frac{y_{i+1}-y_i}{h_i}=\eta_i f(x_i,y_i)+\beta_i f(x_i+\gamma_i h_i, y_i+\delta_i h_i),........(1)$$ where $\beta_i, \gamma_i, \delta_i, \eta_i$ are ...
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39 views

Using Adams-Bashforth-Moulton Predictor Corrector with Adaptive Step-size

I'm investigating the behaviour of predictor-corrector methods to numerically give approximations to the Initial Value Problem. I have currently implemented a Forward-Backward euler method like so, $...
0
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1answer
21 views

Acquiring a function from a butcher table? (Runge-Kutta)

When I have this butcher table: \begin{array} {c|cccc} 0\\ c & c\\ \hline & (1-d) &d \end{array} I got this equation: $\phi(x,y,h) = (1-d)*f(x,y) + d*f(x+ch, y+ch*f(x,y))$ Now I need ...
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2answers
56 views

At which maximum step size is the method stable for real $\lambda$?

I have this Runge-Kutta method: \begin{array} {c|cccc} 0\\ \frac{1}{3} & \frac{1}{3}\\ \frac{2}{3} &0 &\frac{2}{3} \\ \hline & \frac{1}{4} &0 &\frac{3}{4} \end{array} I have ...
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14 views

Not sure what this theorem outputs? Related to proofs/orders/runge-kutta.

I have this theorem: Let $D ⊂ R^m$ be convex, and let $f ∈ C^2 (D, R^n)$. For each $a ∈ D$, there exists $R_a: D → R^n$ such that $$f(x) = f(a) + \frac{\partial \textbf{f}}{\partial \textbf{x}}\...
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32 views

Computing the region of stability for a Runge-Kutta (Order 3) method?

I can't find a method on how to do this anywhere, so I'm having to ask. Thanks for any help or method you can give. I have a butcher tableau that looks like this: \begin{array} {c|cccc} 0\\ \frac{1}{...
0
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1answer
42 views

Calculation using Euler's method

Given y' = 1 - 2x - 3y, starting condition y(4)=5 and h = 1/2 I am asked to estimate by hand the value for y(5). My question is, if my staring value are as follows: ...
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29 views

Which equations are best suited for evaluation of numerical algorithms for solving of ordinary differential equations

In my advanced seminar, I have to implement and evaluate numerical methods for solving of ordinary differential equations. I need to run the implemented methods on the Raspberry Pi Board and evaluate ...
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1answer
27 views

Linearity of discretized ODE

I'm currently reading a section of a book on an implicit Ruge-Kutta method, and the following is written: We start, say, with a diagonally implicit Runge-Kutta method $$k_i = hf\left(y_0+\sum\...
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2answers
84 views

Bring ODE into a suitable form to solve it with Runge-Kutta steps

Can anyone please help me understand, how I should bring this ODE y'' + y = sin(t) with initial conditions y(0) = 100, y'(0) = 5 into a Runge-Kutta-Form? I tried to solve this equation, the ...
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1answer
70 views

Numerical scheme for coupled PDEs

I am trying to solve the three coupled PDEs; $\frac{\partial{Q}}{\partial{t}} = -RaPra^2\theta - Pra^2Q + Pr\frac{\partial^2{Q}}{\partial{z}^2}, \ \ \ \ \ \ \ \ \ (1)$ $\frac{\partial{\theta}}{\...
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23 views

Applying the method of lines to a partial differential equation and using Runge-kutta method

By method of lines I converted the PDE u_t=u_{xx}, with the initial and boundary conditions ...
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1answer
47 views

Empirical error proof Runge-Kutta algorithm when not knowing exact solution

I'm implementing a RK solver for calculating the solution to the Lorenz system: \begin{equation} \begin{cases} x'(t) = \sigma(y-x) \\ y'(t) = rx-y-z \\ z'(t) = xy-bz \end{cases} \end{equation} The ...
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66 views

Calculating the constants for Runge-Kutta order 4 in other form

I know why Runge-Kutta order 4 can be written in the below form I guess. But I don't know how I should go about to calculate the constants required. Runge-Kutta order 4 can also be written in the ...
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33 views

Step size updating scheme adaptive embedded RK methods

If I have a RK method $y$ of order $p$ and a RK method $z$ of order $p-1$ I have read I can estimate the local error as $r_{n+1} = y_{n+1} - z_{n+1}$. First of all I don't see how this estimates the ...
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22 views

Positive definiteness of a Runge-Kutta method

A Runge-Kutta method can be characterized by the $s \times s$ matrix $A$ and the $s$-element column vectors $\mathbf{b}$ and $\mathbf{c}$. In this paper, a special type of Runge-Kutta method is ...
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1answer
20 views

Determine conditions on parameters (for consistency) on RK method $y_{n+1} = y_n + ha_1f(t_n,y_n) + ha_2f(t_n + b_1h, y_n + b_2hf(t_n,y_n))$

I'm asked to find the conditions on the coefficients $a_1,a_2,b_1,b_2$ in the RK method $$y_{n+1} = y_n + ha_1f(t_n,y_n) + ha_2f(t_n + b_1h, y_n + b_2hf(t_n,y_n))$$ such that is consistent of (a) ...
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1answer
24 views

Show that $H(v, y) = \frac{v^2}{2} - F(y)$ is a first integral

$H(v, y) = \frac{v^2}{2} - F(y)$ is a hamiltonian of $y' = v, v' = f(y)$ Linear first integrals are of the form $I(x) = b^Tx + c$ where $b \in ℝ^d $ and $ c \in ℝ$ Quadratic first integrals are of ...
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2answers
27 views

Runge Kutta Method for 2nd ODE [closed]

Given the equation $$ y''=c\cdot(1+(y')^2)^{1/2} ~~where~~ c=0.053. $$ Putting this in system form, I get \begin{align} y'&=z \\ y''&=c⋅(1+z^2)^{1/2} \end{align} I am to use 4th Order Runge-...
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0answers
120 views

Advantage of multi steps ODE methods over single step methods

I wanna know what's the advantage of multi step ODE methods such as Adams-Bashforth over ordinary single step methods such Runge–Kutta, accuracy/time wise.
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1answer
162 views

Runge-Kutta fourth order with negative stepsize

I am solving an ODE using the Runge-Kutta method 4th order and the integration is backward i.e step size ($h$) is negative. All the references that I have seen consider the positive step size. Is it ...
0
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1answer
31 views

Iteration algorithm for finding better approximation in Shooting method for solving BVP

Good day, everyone. Basically , the problem I am given is to solve the system of differential equations with 4 equations, and I have two initial values, and two boundary conditions. Following the ...
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1answer
67 views

Runge-Kutta method for higher-order differential equations

I am studying Numerical Analysis with the book of Richard L.Burden. A question which I'm struggling with right now is following. Transform the second-order initial-value problem $y'' - 2y' + ...
0
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1answer
44 views

Runge Kutta Order 1 and Stage 2

I am looking for examples of Runge-Kutta method with order 1 and stage 2, Then if it is possible, can you exemplify order 1 and stage 3 RK? I am really confused with the stage and order in runge kutta....
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1answer
105 views

solving third order nonlinear differential equation [closed]

Can someone help me in solving : $$ y'''(x)+2\,y(x)y''(x)=0 $$ using Runge-Kutta method? I'm not sure how to linearize it. Thanks in advance
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140 views

Stochastic Differential Equation and Runge-Kutta

I have to solve the Black-Scholes equation, $\textrm{d}X\left(t\right)=\lambda X\left(t\right)\textrm{d}t+\mu X\left(t\right)\textrm{d}W\left(t\right),$ by making use of a RK method (in Python). ...
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0answers
46 views

A matlab implementation for Runge-Kutta 4th order question [closed]

$$ dx/dt= 1-y \\ dy/dt= x^2-y^2 $$ $t=[0,5]$, $h=0.1$, $x(0)= -1.2$, $y(0)= 0$, Am I right about writing code ? Can you check it out ? ...
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1answer
45 views

How can I derive the dense output of ode45?

I'm currently looking at the implementation of the Dormand-Prince 5(4) Runge-Kutta algorithm (also known as Dopr5, or ode45) in ...
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3answers
61 views

Does Runge Kutta methods run well with variable $h$?

When using Runge Kutta methods for real time simulations there is a problem with the constant step length $h$ since the operating systems often interrupt the simulation and the main loop therefore has ...
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1answer
298 views

Solving a matrix differential equation using Runge-Kutta methods

I could not find a control systems forum on stack exchange and so I am doing this here. Is it possible to solve the state space variable form of a system $\dot{x}=A\,x + B\,u$ using any order of Runge-...
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0answers
43 views

How do you choose a suitable integration method?

I'm trying to solve a system of 20 coupled differential equations through computing. These equations exhibit strong oscillatory solutions over millions of years. I have been reading around, however I ...
0
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1answer
58 views

IVP differential equation with Euler Method

Okay the question is: "Compute numerical approximations to the above IVP on the interval $[t_0,te]=[0,1]$. Use step sizes of $h = 0.1, 0.01, 0.001$ and display in a table the results $y_j$ and the ...
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2answers
176 views

MATLAB Error Implementing Fourth Order Adams-Moulton Method

Trying to implement the fourth order AM method in MATLAB using fourth order RK to get the first four starting values. Code is as follows: ...
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0answers
47 views

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 ...
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2answers
45 views

numerical solution for discountinuous differential equations

Let a system of differential equations has the following constraint: $$\frac{d^2x}{dt^2}=f_x(t, x,\frac{dx}{dt})$$ $$\frac{d^2y}{dt^2}=f_y(t, y,\frac{dy}{dt})$$ if $$\frac{dx}{dt} > \frac{dy}{dt}$$ ...
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1answer
53 views

Runge-Kutta 4th order iteration for $\ddot{z} = -\frac{z}{|z|}$

I defined a potential function as $$ U(z) = |z| $$ this encodes a pressure field (or kind of). Assuming I have a particle of mass $m = 1$ I got the following motion equation $$ \ddot{z} = -\frac{z}{...
2
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1answer
271 views

Is Backward-Euler method considered the same as Runge Kutta $2^{\text{nd}}$ order method?

I have a book that quotes: Euler's method, Modified Euler's method and Runge's method are Runge-Kutta methods of first, second and third order respectively. The fourth-order Runge-Kutta method ...
2
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99 views

$1D$ heat equation with $RK4$ method

I know that $Runge-Kutta$ method is a powerful method for ODE. So far I would like to solve $1D$ heat equation with Matlab. My problem is that $k$ (thermal conductivity) depends on temperature. $1D$ ...
3
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2answers
129 views

$4^{\text{th}}$ order Runge-Kutta method

I would like to know the motivation behind the choice of numbers or coefficients in front of $k_1$, $k_2$, $k_3$ and $k_4$ in $4^{\text{th}}$ order Runge-Kutta method. There are many choices of the ...
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0answers
46 views

Rosenbroke - Kaps-Rentrop method

Hi everybody I would like to implement a Rosenbroke methods, I found the equations in this pdf eq 9 but are not so clear for me may somebody explain me and make a bit of order of these equations ?
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1answer
100 views

How to get the Radau 5th order equations give the Butcher tableau

I'm don't well understand how from the tableau of Butcher I can find the equations needed for implement the method, In particular I want be able to know the equations for Radau method 4th and 5th ...
0
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1answer
131 views

how to write the Radau 2nd order methods (Butcher’s table)

I would like to solve the Robertson problem you can find the system here this is a stiff system of ODE which require an implicit high (more than one) order solver. In particular the RADAU IIA was used ...
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0answers
55 views

Difficult to understand a method for stiff ODE

I really didn't find a well explained step necessary to implement the (implicit Runge Kutta) Rosenbrock method ... I found this paper : here you can visulize eq. 9 but is not so clear, first of all ...