Questions tagged [runge-kutta]

The Runge–Kutta methods are a set of numerical methods for ordinary differential equations for the approximation of their solutions.

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Jacobian matrix cutoff in ODE solver

I am studying an implementation of a semi-implicit Runge Kutta method of 3. order (SIRK3) from the book by Villadsen & Michelsen (1978), Solution of differential equation models by polynomial ...
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150 views

N-body problem with differents solvers (RK2, RK4, Euler symplectic, Stormer-Verlet) : planets drift to infinity

I'm trying to write an integrator for the 2 and 3-body problem. I choose to start from a generalisation to N-body problem so I can just pass my bodies to the same integrator in the two cases. I'm ...
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61 views

Divergence on wave equation simulation

I'm currenly working on my own PDE solver for non-linear simulations in python. I've done succesfully simulations for KdV and Fisher's equation, but now I'm playing with second order derivatives in ...
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147 views

Perturbation problem using Runge-Kutta 4

I'm trying to evaluate the perturbations magnitude between 2 body orbiting a central one in three dimensions. In order to do this I need to have an estimate of the error, which I did using Richardson ...
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373 views

Special-case Runge-Kutta methods to exploit structure in linear ODE?

I am interested in numerical solutions of a linear, time-dependent ODE of the form $$ \dot y = A(t)y - Ry, $$ A good model is the following problem in $\mathbb R^2$: $$ A(t) = \begin{bmatrix}0 & -\...
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Using the BDF and RK4 methods to solve this coupled system of ODEs in C++

I'm trying to solve a system of ODEs using the BDF order 4 method. I find the first 3 points using RK4, then for the implicit part of the BDF, I use Newton-Raphson iteration. Unfortunately my solution ...
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98 views

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 $...
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59 views

Python evaluating a second order ODE with RK4

Pasted below is my python code. It is a 4th order runge kutta that evaluates the 2nd order ode: y'' +4y'+2y=0 with initial conditions y(0)=1, y'(0)=3. I need help fixing it. When I run my code, my ...
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141 views

How to get a more accurate cancelation

I shall try to get to the point, so let me know if there is something left and you need more details. I am solving a couple of equations that are not coupled explicitly, but their corresponding ...
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1answer
269 views

DOP853 integration method is missing (SciPy)

I was checking some integration methods provided by SciPy, in which the DOP853 should be included according to the webpage (https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate....
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108 views

Coupled second-order differential equations using runge kutta 45

As a little summer project I have tried to make a ballistic calculator for when I play football, (following an example from a book), just to learn some numerical methods while doing so. My problem is ...
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72 views

How is the final result calculated in RK-Dopri(4,5)?

I have found a toy implementation of RK-Dopri(4,5), written in Python. I am concerned however, about line 118: y = y + h * (b1*K1+b3*K3+b4*K4+b5*K5+b6*K6) Has the ...
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RK4-method starts oscillating above certain input parameters

I am trying to solve an equation of the following type $$\partial_zE(z)=-c_0J$$ with $$J=c_1\beta E^3(z)$$ using the boost::odeint-framework and a fixed time stepper, with $c_0$, $c_1$ and $\beta$ ...
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113 views

Method of Lines Runge-Kutta nonlinear stability and behavior

I have a system of 4 nonlinear 1st-order PDEs. I want to solve them numerically by method of lines, first discretizing space. This leads to the system of $N\times 4$ coupled ODEs. $$ \mathbf{u}_{i} =...
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125 views

Two RK4 method in one program

I want to solve this integral using RK4 by coding in Fortran: $$R=∫1/a(t) dt → dR/dt=1/a(t) =f(t)$$ Initial point: t=0 (or a=0.001) and R=0 And I have to get a(t) by solving another ...
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68 views

Time independent Runge Kutta integration of SDE

I am trying to compare the result of numerical integration of time independent Runge_Kutta, github page for stochastic differential equations with the analytical solution. True answer match the ...
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151 views

Runge-Kutta method for an ODE with initial value which is root of denominator

I wrote a code in Fortran to solve this differential equation using RK4 method: $$ \frac{dy}{dx}=A\sqrt{\frac{B}{y}+\frac{C}{y^2}} $$ $A$, $B$, and $C$ are some known constants. The problem is that ...
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Changing variables in integral to avoid infinity

I want to write a code in Fortran to solve this integral numerically: $$\int_{1095}^\infty \frac{dx}{x\sqrt{(x+644.153)(4.17 \cdot 10^{-5} x+0.145)}}$$ What is the best method for it? I tried Monte-...
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244 views

Solving nonlinear pendulum using Runge-Kutta 4 for smaller steps

I am trying to solve nonlinear pendulum using 4th order Runge-Kutta method for limits between a=0.0 to b=110 seconds and simulated the results to observe the pendulum movement. But when I increase the ...
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1answer
60 views

Adaptive Runge-Kutta for Stochastic (Projected) Gross-Pitaevskii Equation

I am using the XMDS library for solving the stochastic (projected) Gross-Pitaevskii equation $$i \hbar \partial \Phi\left(\mathbf{r},t\right)_t=\hat{\mathcal{P}}\left\{(1-i \gamma)\left(\hat{H}_{\...
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1answer
178 views

Solving ODEs with nonlinear constraints

I'm trying to solve an ODE problem. Let's say $\mathbf{x}(t)$ represents the position of a particle at time $t$, and $\mathbf{u}(\mathbf{x},t)$ is a velocity field defined in Cartesian coordinates on ...
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67 views

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 ...
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259 views

Solving coupled ODEs using Runge-Kutta method

I want to solve the following sets of $n$ coupled equations. Initial values of $x_{n}(t)$ and $p_{n}(t)$ are specified. The problem is, I have an 1D lattice where every particle is bound with ...
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81 views

Type of Rosenbrock method by its coefficients

A Fortran code that solves stiff PDE systems contains the following arrays of Rosenbrock-Wanner method coefficients: ...
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962 views

Numerically solving a partial differential equation in python with Runge Kutta 4

I'm supposed to solve the following partial differential equation in python using Runge-Kutta 4 method in time. $$ \frac{\partial}{\partial t}v(y,t)=Lv(t,y) $$ where $L$ is the following linear ...
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340 views

The Formula of Explicit Runge-Kutta Fourteen order

I need an explicit Runge-Kutta 14th order formula. If you know about some reference that discusses at least 10th order (or higher, since I'm looking for the 14th) of Runge-Kutta and there is ...
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1answer
302 views

Solving differential equation in Python with discretized variable coefficients

I am trying to solve a differential equation with discretized variable coefficients which are calculated from a time serie. In this case the Runge-Kutta step size is fixed by the frequency in the time ...
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Runge-Kutta in the presence of an attractor

Suppose you are solving a system of equations numerically that possesses an attractor (no matter the initial conditions set, all the different solutions will approach a specific set of values that ...
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158 views

What exactly is the cause(s) of blow-up for too-large step size in a method like RK4?

I have been working on creating a few home-made numerical methods, and I am using them to visualize text-book problems from my Strogatz dynamics textbook. It feels like a good way to learn numerical ...
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346 views

Runge-Kutta fourth order method. Integrating backwards

I am using a Runge-Kutta fourth order method to solve numerically the usual equation of motion of a background scalar field in curved spacetime with a quartic potential: $\phi^{''}=-3\left(1+\frac{H^{...
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111 views

How to Break Coupled ODEs down to first order for Runge-Kutta

My question might seem a bit simple. I am trying to solve a system of ODEs using Runge-Kutta method. I am having difficulty breaking down the equations into a system of first order ones required ...
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171 views

Runge-Kutta for PID and system in separate calculations without filter

I need to calculate a closed-loop system in Python; specifically, obtain the PID response and then use the output to obtain the system response sample-by-sample with my own loop. For this, I am ...
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2answers
68 views

Verifying convergence of a stationary solution to a PDE with the Runge-Kutta method

I am numerically solving a nonlinear wave PDE using the Runge-Kutta method, and I know the solution I am looking for is constant in time, but I do not know the solution. What is a good way of ...
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65 views

Using Kutta Merson on NLS

I'm trying to use the Kutta-Merson to get the same results as in the book Solitons, Nonlinear Evolution Equations and Inverse Scattering - M. J. Ablowitz - pg 140 The author propose using the Kutta-...
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1answer
273 views

Langevin equation in 4th order Runge-Kutta

I'm trying to figure out how to translate a piece of code from Velocity Verlet to Runge-Kutta, while treating the time step dependence of the thermal noise correctly. The Langevin equation for my ...
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61 views

Reverse automatic differentiation and integration

In Symplectic Runge-Kutta schemes for adjoint equations, automatic differentiation, optimal control and more Sanz Serna writes: It is well known that the reverse mode of differentiation implies ...
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Computing Trajectory Equations of Kerr Geodesics

I want to numerically solve the trajectory equations of a Kerr geodesic given by wikipedia in Matlab. The trajectories look like: I implemented the equations and solved it with the standard Runge-...
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345 views

Is there any explicit symplectic Runge-Kutta method?

As far as I know, all the symplectic Runge-Kutta methods are implicit which need to solve non-linear equations during the calculation. Is there any explicit method? If not, why?
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901 views

Dormand–Prince 5(4): How to update the stepsize and make accept/reject decision?

https://en.wikipedia.org/wiki/Dormand–Prince_method I want to implement the Dormand-Prince 4(5) version to solve Initial Value problems. Using regular notation I have $A$ matrix and the $c,b,\hat{b}$ ...
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475 views

Is there a database/website with Butcher tableaus?

I have started investigating in mostly Runge Kutta and Runge Kutta Nyström methods and there one of the only differences between the methods of the same type is their Butcher tableu. For the most ...
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How to implement adaptive step size Runge-Kutta Cash-Karp?

Trying to implement an adaptive step size Runge-Kutta Cash-Karp but failing with this error: ...
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1answer
211 views

How to obtain and form a 1st order differential equation for leapfrog integration from second order one in this example of coulomb drag

I am currently doing a computational physics homework which asked us to use leapfrog to give the relations between timevelocities and time-distance of these two objects. The full question is as ...
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234 views

Finding second excited state of Schrödinger equation with secant Runge Kutta method

In our assignment, we are required to find the energies of the ground state and the first two excited states of the Schrödinger equation in a harmonic potential: $$V = \frac{50 x^2}{(10^{-11})^2}\, .$...
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103 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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TVD for temporal dicretisation

I have come across schemes where TVD (with flux limiters) is used for spatial discretisation along with Runge-kutta for Temporal discretisation. Can TVD be used for Temporal discretisation? If so ...
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1answer
51 views

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 ...
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49 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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338 views

How well do explicit Runge-Kutta “tableau” methods compare to the state of the art ODE solvers and when do they fail?

How well do explicit Runge-Kutta "tableau" methods compare to the state of the art ODE solvers and when do they fail? I've been reading Butcher's ODE book and he does a good job at introducing ...
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1answer
139 views

Euler Method Instability. Why?

I am currently enjoying writing computational codes as a hobby. Right now I've worked out an Euler method and results are pretty good with up to $x=1$. Over $x=1$, instability starts to kick in. May I ...
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1answer
696 views

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(...