# Tag Info

Accepted

### What does "symplectic" mean in reference to numerical integrators, and does SciPy's odeint use them?

Let me start off with corrections. No, odeint doesn't have any symplectic integrators. No, symplectic integration doesn't mean conservation of energy. What does ...
• 11.5k
Accepted

### Which Runge-Kutta method is more accurate: Dormand-Prince or Cash-Karp?

Since I just finished optimizing a lot of them in a software, DifferentialEquations.jl, I decided to just lay out a comparison of the main Order 4/5 methods. The Fehlberg method was left out because ...
• 11.5k
Accepted

### What is the state of the art in solving stiff initial value problems?

So there is a ton to say about this, and we will actually be putting a paper out that tries to summarize it a bit, but let me narrow it down to something that can be put into a quick StackOverflow ...
• 11.5k
Accepted

### Why are higher-order Runge–Kutta methods not used more often?

There are thousands of papers and hundreds of codes out there using Runge-Kutta methods of fifth order or higher. Note that the most commonly used explicit integrator in MATLAB is ODE45, which ...
• 16.2k

### What does "symplectic" mean in reference to numerical integrators, and does SciPy's odeint use them?

To complement Chris Rackauckas answer, to state some of the mathematical nonsense as well as some stuff you almost certainly know, a dynamical system is Hamiltonian if there is a description with ...
• 2,189
Accepted

### Why is RK45 used as the "default" method for non-stiff ODEs rather than a multistep one?

First, let's establish that they are a good choice. The SciMLBenchmarks are probably the most comprehensive that there are as of right now for modern methods. This uses the vast number of methods ...
• 11.5k
Accepted

### Constructing explicit Runge Kutta methods of order 9 and higher

Bounds That is still true. In Butcher's book, page 196, it says the following: In a 1985 paper, Butcher showed that you need 11 stages to get order 8, and this is sharp. For order 10, Hairer derived ...
• 16.2k

### What's the state of the art in parallel ODE methods?

Although this post is now two years old, in case someone stumbles across it, let me give a brief update: Martin Gander recently wrote a nice review article, that gives a historical perspective on the ...
• 1,198

### Why does the numerical solution of an ODE move away from an unstable equilibrium?

Note that $\pi/2$ is represented in double precision format in a way that is not exactly equal to $\pi/2$. It's only accurate to about 15 digits. Thus you're starting every so slightly away from the ...
• 17.6k

### Why are higher-order Runge–Kutta methods not used more often?

The Benchmark Setup In the Julia software DifferentialEquations.jl we implemented plenty of higher order methods, including the Feagin methods. You can see it in our list of methods, and then there ...
• 11.5k
Accepted

### Why does the numerical solution of an ODE move away from an unstable equilibrium?

I think the two main points have already been made by Brian and Ertxiem: your initial value is an unstable equilibrium and the fact that your numerical computations are never really exact provides the ...
• 1,198

### Options for solving ODE systems on GPUs?

DifferentialEquations.jl library is a library for a high level language (Julia) which has tools for automatically transforming the ODE system to an optimized version for parallel solution on GPUs. ...
• 11.5k