Questions tagged [runge-kutta-methods]

For questions about the family of Runge–Kutta methods and their application in numerical methods.

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Convert the pendulum differential equations of a second order into a first order system [closed]

I have this system of two differential equations of a second order. I got them from the Euler-Lagrange equations of double pendulum. I need to solve this using the Runge-Kutta numerical method, but my ...
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How to deal with discontinuities in the Runge-Kutta algorithm?

I am trying to solve the following set of differential equations for plotting particle trajectories (using 4th order Runge-Kutta method): $$\frac{dr}{dt}=-\Delta\left[\frac{2}{r}\left(1+\frac{a^2}{r^2}...
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Embedding methods into implicit Runge-Kutta schemes

I was reading through Hairer's book for stiff ODEs, and while mainly looking at the theory, I ended up reading chapter IV.8 on implementation details, just to write a toy program to see things in ...
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Prove that the region of absolute stability for explicit Runge-Kutta methods is bounded

That's what I've got: We know, that the region of absolute stability are those $z \in \mathbb{C}$, for which $|R(z)| < 1$, where $R(z)$ is the stability function. For explicit Runge-Kutta methods, ...
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If a numerical method for ODE is explicit, does that mean that it's got exactly one unique solution?

I should prove, that the following RK method: $$x_{k+1} = x_k + \frac{h}{6} (K_1 + 2 K_2 + 2 K_3 + K_4)$$ for: $$K_1 = f(t_k, x_k)$$ $$K_2 = f(t_k + \frac{1}{2} h, x_k + \frac{1}{2} h K_1)$$ $$K_3 = f(...
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Theory behind the Runge-Kutta method

RK4 is: $$\begin{aligned}k_1&=hf(x_n, y_n)\\ k_2&=hf(x_n+h/2. y_n+k_1/2)\\ k_3&=hf(x_n+h/2, y_n+k_2/2)\\ k_4&=hf(x_n+h, y_n+k_3)\\ y_{n+1}&=y_n+(k_1+2k_2+2k_3+k_4)/6\end{aligned}$$ ...
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Is there a generalized formula for solving systems of ODEs using 4th order Runge-Kutta?

As far as I know, we numerically solve any system by reducing it to ODEs and somehow we manage to wire the new system into the RK algorithm. I have seen solutions where there are formulas to use the ...
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Applying a Short Force Impulse on my Object During Runge-Kutta Integration

I'm fairly familiar with Runge-Kutta ODE solvers, but recently I have begun to try to add collision to my soft-body and hard-body objects. I believe the safest and most general way to model this would ...
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Is there a formula for the Butcher tableau for the implicit Runge Kutta method for an arbitrary number of stages?

Usually the Runge Kutta method uses 4 stages. What if I want to use an arbitrary number of stages like 300 stages, which is possible in some domains (e.g. with physics informed neural networks). Is ...
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Solving 2nd order Diff Eqn by RK4 method

I am trying to solve 2nd order differential equation for a harmonic oscillator, $$ y''+2 \beta y'+ \omega^2 y =0, $$ using RK4 method in Fortran for different values of beta n omega, program is ...
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Study of stable oscillations of an inverted pendulum

A rigid rod of length L, supported at one end by a frictionless hinge, connected to an electric motor, through which rapid vibrations are forced hinge in the vertical direction: $$S = A\sin(\omega t)$$...
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Generalized Runge-Kutta Method

I am trying to understand analytical theory of RK method. I am following the book A first course in Numerical Analysis by Ralston and Rabinowitz(2nd Edition) Page 225. We start with the solution of $\...
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Doubt regarding solving a "system of" second order differential equations using 4th order Runge-Kutta method

To solve a single 2nd order differential equation using the 4th order Runge-Kutta method, it is desirable to write the equation as two coupled first order differential equations. Two such coupled 1st ...
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How to find the Butcher Tableau of this Runge-Kutta method? [closed]

I have this equation $y_{n+1} = y_n + hf(y_n + \frac{h}{2} f(y_n))$, I am thinking of letting $Y_i = y_n + \frac{h}{2}f(y_n)$ and then having it written as $Y_1 = y_n + \frac{h}{2}f(Y_1)$, does anyone ...
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Implicit Euler looks different from Runge-Kutta implicit Euler

In theory Implicit (Backward) Euler should be a Runge-Kutta method with tableu given below, however I find that the standard Backward Euler formula and the Runge-Kutta one differ. $$ {\begin{array}{c|...
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Converting Method to Butcher tableau

Suppose that a method is given by $$y_{n+2/3} = y_n + h((1/3) f(y_{n+2/3}) + (1/3) f(y_{n}))$$ and $$y_{n+1} = y_n + h((3/4) f(y_{n+2/3}) + (1/4) f(y_{n})).$$ I am trying to obtain the Butcher tableau ...
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Understanding Runge-kutta and Linear multistep method

In the Numerical Analysis by Richard L. Burden, section $5.4$ was told that The first step in deriving a Runge-Kutta method is to determine values for $a_{1}, \alpha_{1}$, and $\beta_{1}$ with the ...
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Runge–Kutta integration for quaternion kinematrics

How can I use the Runge-Kutta integration method for quaternions Kinematics (https://arxiv.org/pdf/1711.02508.pdf) Incremental rotation $\Delta \theta = \omega_n \Delta t$ $ Exp(\omega \Delta t) = q\{...
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Scaling variable in ODE system

Let's say we have an ODEs system of the type: \begin{eqnarray} \frac{\partial \vec{y}(t)}{\partial t}=f(t,\vec{y}(t),\partial_{t}\vec{y}(t) ), \end{eqnarray} with $t\in[0,\infty)$. We want now to ...
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Is $\dot u_h(t_0 + t h) = \sum_{j = 1 \dots s} \dot u_h(t_0 + c_j h) L_j(t)$ a true interpolation formula?

I'm studying collocation mathods to solve ODEs and I got stuck on the proof of the Guillou-Soule theorem which proves that a collocation method is a particular kind of implicit RK method under some ...
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How to implement implicit Runge-kutta (RK-4) method for a system of non-linear differential equation?

I am trying to solve a set of non-linear ode's using an implicit RK-4 method. I have already solved this problem using a backward euler's method. $$ \frac{dy_i}{dt} = a_i*y_{i-1} - b_i*y_i + c_i*y_{i+...
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Runge-Kutta method using taylor series

I was studying Runge-Kutta method for solving $\frac{dy}{dt}=f(t,y)$, but I don't understand how can I write $$y_{n+1}=y_n+ak_1+bk_2$$ using the Teylor series where $k_1=\Delta t f(t_n,y_n)$ and $k_2=\...
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Runge-Kutta method for state space with input

I need to implement the 4th order Runge-Kutta method to propagate an affine non linear system. My doubt is in regards on how the input should be treated when calculating the k parameters. To be more ...
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Question over Nodes of Runge-Kutta methods

I have (hopefully) an easy question. Suppose I have the following system (derived from an integration) $I'(t)= f(t), \, t \in [0,\pi]\\ I(0)=0$ My goal is to find $I(\pi)= \int_{0}^{\pi}f(t) \text{d}t$...
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Pollution Problem in a room: find the time when the concentration of carbon monoxide in the room reaches $0.01\%$

A human cabin in a spaceship contains $4500\,m^3$ of air, initially free of carbon monoxide. Starting at time $t = 0$, smoke from one of the machinery room canisters containing $4\%$ carbon monoxide ...
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NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS. [closed]

Here is a problem, this is in my compulsory list of homework, I'm trying to solve but it seems very hard for me, I've tried Taylor expansion, considered some cases of the value c,... Can you give me ...
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Areas and adaptive-step Runge-Kutta methods

I have been using standard, non-adaptive symplectic integrators in my work for many years. This kind of integrator requires the user to provide a time-step $\Delta t$, which is kept fixed during ...
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Periodic boundary condition and Runge-Kutta method of order 4 used in solving Partial Differential Equation(1D Wave packet equation).

I have to code for the problem of solving 1D wave packet propagation with equation given as: $$\frac{\partial u}{\partial t} + c\frac{\partial u}{\partial x} = 0$$ The initial solution is given as: u(...
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Solving Two Body Problem With Second Order Runge-Kutta

I read a paper which want to solve two body problem with Second Order Runge-Kutta (the paper want to find optimum weight of Runge-Kutta with ANN). Here it is the two body problem. \begin{align*} x"=\...
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Runge-Kutta time propagation with large stochastic error

I have to solve a differential equation of the type: $$ \frac{d y}{dt} = f(t,y) $$ $$ y(t=0) = y_0 $$ In general one would solve it with some high order Runge-Kutta method... if one could compute $f(t,...
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Where to find a comprehensive review of Butcher's tableau and dense step expressions for most used RK methods?

I need to implement several RK methods with adaptive timestep. I am however wasting a lot of time in finding consistently Butcher's tableau AND the expressions to evaluate the solution on dense steps. ...
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Absolute stability of numerical methods for ODEs

I've troubles understanding the meaning of region of absolute stability for numerical methods for ODEs. I know that we can restric the study of stability of a certain method to the case of the test ...
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How to implement Runge-Kutta 4 method to calculate the trajectory of a particle in a vector field?

I'm currently trying to approximate the trajectory of a particle inside a 2D vector field. The particle has an initial position $P = (a,b)$, an initial velocity vector $\vec{V} = (p,q)$, and a given ...
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Numerical integration of differential equation in state-space form

I'm using numerical integration methods like Explicit/Implicit Euler, Runge-Kutta to solve a system of linear ordinary differential equations in state-space representation $\dot{x}=A\,x + B\,u$. I ...
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Discretization of ODE system

I am fairly new to the discretization of ODE systems (indeed a good reference would be helpful). I have a system of ODEs that basically looks like this $$ \begin{align} \frac{d x(t)}{dt} &= v(...
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Solving non-linear PDE with Runge-Kutta 4th order

I want to solve the following non-linear PDE with Runge Kutta 4th order: $$\partial_t y(t,x)=y(t,x)\partial_x y(t,x)- 3t^2=:f(t,y)$$ The initial conditions $y(t_0,x_j)=:y_0^j$ are given at each ...
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Help with predictor-corrector method

For equation $u_t+au_x=0, a=const$ we have predictor-corrector scheme: $$ \frac{u{}_{m+1/2}^{n+1/2}-\frac{u_{m+1}^n+u_m^n}{2}}{\tau/2}+a\frac{u_{m+1}^n-u_m^n}{h}=0, $$ $$ \frac{u{}_{m}^{n+1}-u_m^n}{\...
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runge kutta 2 in python

I am trying to solve an equation in fluid mechanics using the runge-kutta 2 method, usually it seems quite doable but in this case its with x y and z and i cant seem to make the code. Here is what i ...
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Runge-Kutta 4 with multiple equations and no time dependence

I'd like to believe that I understand the Runge-Kutta 4 method; however, I am having difficulty applying to non-standard cases. Runge-Kutta 4: $$ k_1 = hf(x,t) \\ k_2 = hf(x + \frac{k_1}{2},t + \frac{...
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Deriving the Runge-Kutta formula, repeated derivatives

This is a part of the derivation of a Runge-Kutta method. Consider the equation $$y' = f(t,y)$$ then $$y'' = f_t+fy\cdot y'$$ $$y'''=f_{tt}+f_{ty}f+(f_t+f_yf)f_y+f(f_{ty}+f_{tt}f)$$ where $f_t,f_y$ ...
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Runge Kutta adaptive step size collapses to 0

I am trying to solve numerically a problem in orbital motion, using Runge-Kutta 4 method with adaptive step size. Because energy is the most obvious theoretically conservative number in the motion, I ...
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Stability of Explicit midpoint method

I am trying to determine the stability region of the well known explicit midpoint method $$y_{i+1} = y_i + h f\left( t_i + \frac h 2, \ y_i + \frac h 2 f(t_i, y_i)\right)$$ and after following the ...
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BVP sine-Gordon problem solved with Runge Kutta

We have the following BVP on xgrid [-2,2] and tgrid[0,4] $$ u_{tt} -u_{xx} + \text{sin}(u)=0, u(x,0) = \sin(\pi x)^2e^{-x^2}, u_t(x,0)=\sin(\pi x)^4e^{-x^2}, 0=u(-2,t)=u(2,t) $$ and we have to solve ...
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Applying numerical schemes like Runge Kutta to hyperbolic PDEs where the partial derivative of one variable is a function of another variable

If anyone can help with how to begin in this case, it would be a great help. Imagine if we have a 1D hyperbolic system of equations like $\frac{\partial U}{\partial t}=-i\omega\zeta$ and $\frac{\...
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Asymmetrical step size to apply Richardson Extrapolation to improve Runge-Kutta order 2 solution

I'm trying to solve a series of problems related to approximations of ODEs with Runge-Kutta that have their approximation to values improved by using the Richardson Extrapolation. Some of these ...
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Is 2nd derivative RK4 method better than classic Runge-Kutta 4?

We have a Newtonian physics simulation with a state matrix containing the instantaneous velocities, angular velocities, positions and orientations of an object ; and we apply classical Runge-Kutta 4 ...
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Showing A-stability for Runge-Kutta-method

For $0\leq \theta \leq 1$ we have given the method $$ x_n-x_{n-1}=h(\theta f(x_,t_n) + (1-\theta)f(x_{n-1},t_{n-1})) $$ Now I should formulate this as a Rune-Kutta-Method and show that it is A-stable ...
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The constrains of Butcher tableau

Related posts(but did not answer the question): deriving an implicit Runge Kutta method from its Butcher tableau Understanding the Butcher tableau of implicit midpoint method butcher tableau for ...
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Implicit Runge-Kutta-methods alternative formulation

Implicit Runge-Kutta-methods with $s$ stepts and with constant lenght $h$ for $$ x'(t)=f(x(t),t), \quad x(t_0)=x_0 $$ have the form $$ X_i=f(x_{n-1}+h \sum_{j=1}^s a_{ij}X_j,t_{n-1}+hc_i), \quad i=1,.....
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Proof of Radau IA method being of order 3

I am trying to proof directly that the Radau IA method \begin{align} k_1 &= f\left(t_0, x_o+h\left(\frac14 k_1 - \frac14 k_2\right)\right) \\ k_2 &= f\left(t_0+\frac23 h, x_o+h\left(\frac14 ...
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