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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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Runge-Kutta 4th order for 4 coupled first order differential equation [closed]

I have to solve 4 coupled first order differential equations for $f(t)$ ,$g(t)$, $h(t)$ and $w(t)$ witch are only functions of $t$ , but for every reference link a function of 3 variables is assumed (...
gdlm's user avatar
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Is there a relatively simple way to extract the Jacobian from a Runge-Kutta 4/5 integrator?

I have a RKF45 numerical integrator that simulates polymerization of proteins using CUDA. It does so by tracking the populations of discrete length polymers, e.g. monomers, dimers, trimers, etc. all ...
Hair of Slytherin's user avatar
5 votes
2 answers
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Machine precision and local error

I'm working with an RKF45 integrator that I have programmed using CUDA C++ on my GPU and am pondering a few questions as I'm trying to track down some issues with my code. I'm using double ...
Hair of Slytherin's user avatar
10 votes
2 answers
2k views

Constructing explicit Runge Kutta methods of order 9 and higher

Some older books I've seen say that the minimum number of stages of an explicit Runge-Kutta method of a specified order is unknown for orders $\geq 9$. Is this still true? What libraries are there ...
Kirill's user avatar
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14 votes
4 answers
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Looking for Runge-Kutta 8th order in C/C++

I would like to use Runge-Kutta 8th order method (89) in a celestial mechanics / astrodynamics application, written in C++, using a Windows machine. Therefore I wonder if anyone knows a good library / ...
James C's user avatar
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3 votes
1 answer
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Algorithm for solving an ODE with time-dependent parameter numerically

Would anyone please explain me what is the mathematical algorithm to solve a IVP system of ODE with a time-dependent parameter. e.g. ...
prashanta_himalay's user avatar
4 votes
3 answers
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4th-order Runge-Kutta method for coupled harmonic oscillator

I’m attempting to write a C program to gather values from a coupled spring system: There is a wall, connected to a mass $m_1$ by a spring, then this mass is connected to a second mass $m_2$ by another ...
Carterini's user avatar
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Why am I getting so much error for my Runge Kutta Fehlberg solver?

My current project is a reprogramming of a protein folding model involving the solution of thousands of ODEs in C++. I've been making some stop and start progress as I'm writing the solver to run ...
Hair of Slytherin's user avatar
1 vote
4 answers
3k views

Best PARALLEL numerical solver of first order differential equation

I have a system of 256 differential equations that I want to solve numerically. The system represents the Liouville equation, which is a first order, linear differential equation with complex numbers. ...
The Quantum Physicist's user avatar
5 votes
2 answers
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Convert an implicit ODE system to an explicit ODE set to use Runge-Kutta

For my series on mechanical systems with Lagrange, I would like to add the double pendulum and a rolling pendulum. I set up the Lagrangian $L$ and solved for the ODE of motions, and got the following: ...
Martin Ueding's user avatar
7 votes
2 answers
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ODE: How to measure stiffness if the Jacobian has zero eigenvalues?

Say you have a system of ODE's where the Jacobian has one zero eigenvalue; what does that tell you about the stiffness of the system? This case doesn't seem to be discussed in the cases I have been ...
trolle3000's user avatar
8 votes
2 answers
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How to decide stability of Runge-Kutta method for non-linear ODE?

I'm working on a parameter study of Duffing's equation $\ddot x + \delta \dot x + \alpha x + \beta x^3 = \gamma \cos{\omega t},$ where $\delta, \alpha, \beta, \gamma$ and $\omega$ are real ...
trolle3000's user avatar
3 votes
0 answers
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Estimating eigenvalues from time-dependent non-linear operator

I have a very sparse non-linear system $N(u) = 0$ that can be solved as a time-dependent ODE, $\frac{du}{dt} = N(u)$, and explicitly integrated until $\frac{du}{dt} = N(u) = 0$, e.g. by forward euler, ...
Aurelius's user avatar
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7 votes
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Intermediate values (interpolation) after Runge-Kutta calculation

I have a numerical ODE simulation that I computed at fixed time step $h$ using a 4-th order Runge-Kutta method (RK4), producing a series of results $(x_1,y_1), (x_2,...
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