82
votes
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 ...
- 12.2k
43
votes
Accepted
Which Runge-Kutta method is more accurate: Dormand-Prince or Cash-Karp?
Since I just finished optimizing a lot of them in 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 it'...
- 12.2k
22
votes
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 ...
- 16.4k
20
votes
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 ...
- 12.2k
18
votes
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 ...
- 12.2k
18
votes
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 ...
- 12.2k
17
votes
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 ...
- 2,229
16
votes
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 ...
- 18.5k
15
votes
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 ...
- 1,238
14
votes
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. ...
- 12.2k
14
votes
Accepted
Forcing an ODE solver to preserve the norm
The best approach is to use an ODE solver that is guaranteed to conserve the norm of the initial condition, i.e., for which $\|y_n\| = \|y_0\|$ for all $n\in\mathbb{N}$. Such solvers exist, and are ...
- 12.1k
13
votes
Why are higher-order Runge–Kutta methods not used more often?
As long as you're using standard double precision floating point arithmetic, very high order methods aren't needed to get a solution with high accuracy in a reasonable number of steps. In practice I ...
- 18.5k
13
votes
Accepted
Comparing Algorithmic complexity, ODE Solvers (Big O)
odeint from the SciPy library defaults to the lsoda integrator described here. However, any simple description of asymptotic ...
- 12.2k
13
votes
Accepted
Why is the central difference method dispersing my solution?
I'll write the equation short as $$\ddot x(t)+c\dot x(t)=a(t,x(t))$$ to separate the "easy" linear parts from the non-linear and forcing terms.
On the first method
The claimed order of the ...
- 5,249
13
votes
Accepted
Are stiffness and instability equivalent?
There are non-stiff problems which are unconditionally unstable with some explicit methods, and conversely there are stiff problems which can be stable with explicit methods. Consider the oscillating ...
- 2,171
12
votes
Accepted
ODEs vs DAE vs ADE?
At least one difference is that in a system of ODEs, all the equations are differential, e.g.:
$$
\dot{x}=f(x,y)\\
\dot{y}=g(x,y)
$$
whereas the definition of DAEs that I'm familiar with includes ...
- 10.9k
11
votes
What is “tolerance” in ODE45 in Matlab?
And, it is my understanding that the 4 and the 5 are for the order of the global and local error, respectively.
Your understanding is wrong.
The local error of a Runge–Kutta method of order $n$ is ...
- 2,022
10
votes
Accepted
Which numerical methods preserve time reversal symmetry?
What one usually wants in this situation is to preserve a discrete analog of time symmetry: namely, if the time discretization is applied to solve first forward and then backward in time, the initial ...
- 16.4k
10
votes
Do there exist low-storage Runge–Kutta methods with an order larger than four?
This area has been fairly well researched, you may check e.g. Ketcheson's review of such methods:
https://doi.org/10.1016/j.jcp.2009.11.006
which does contain some low-storage Runge-Kutta methods ...
- 3,263
10
votes
Accepted
Going back in time in an initial value problem
This is technically still an IVP if you do an appropriate change of variables. Given your time is between $t \in [t^*, 0]$, make a new time variable $\tau = -t$ so that $\tau \in [0, -t^*]$ and you ...
- 3,878
9
votes
Methods for solving $x'=Ax+b$ for small, sparse, singular $A$
Any general-purpose ODE solver should be able to handle this linear coupled system of ODE very easily, for example:
scipy.integrate.ode
CVODE from the Sundials solver suite; it appears to have Python ...
- 11.4k
9
votes
Accepted
4th order Runge-Kutta for $y' = y$
The answer is quite simple. You are already comparing apples and oranges in the first equation. Garbage in, garbage out.
The equation $y'=y$ if written properly is $$dy/dx=y.$$ Do you see it now? To ...
- 206
9
votes
Accepted
Is there any explicit symplectic Runge-Kutta method?
There are explicit, symplectic methods for certain types of Hamiltonian problems. For example, the symplectic Euler method
\begin{align}
p_{n+1} &= p_n - h H_q(p_{n+1}, q_n) \\
q_{n+1} &= ...
- 1,238
9
votes
Accepted
Special-case Runge-Kutta methods to exploit structure in linear ODE?
There are many kinds of RK methods that have extensions to exploit linearity. They all use some form of exponential or Lie Group idea (again exponential) to do so. Thus they generally do some form of ...
- 12.2k
8
votes
ODEs vs DAE vs ADE?
Differential-algebraic equations (DAE) are equations of the form $F(t,x,x')=0$, with the unknown function being $x(t)$. So in a way are generalizations of ODEs. A nice place to start is here. On the ...
- 183
8
votes
Accepted
Computational time not proportional to integration interval in ODE-solver?
Substantially edited, since the original poster changed his equation...
In general, the MATLAB (and Octave) ODE solvers dynamically adjust the step size as needed to maintain an accurate solution. ...
- 18.5k
8
votes
Accepted
Why does LSODA fail to integrate the logistic function?
When you use $r=5$, the initial condition is
$$ x(-10) \approx e^{-50} \approx 1.9\times 10^{-22}. $$
This is much smaller than the machine epsilon, $2\times 10^{-16}$, and it is very likely that ...
- 11.4k
8
votes
Will the numerical solving of the differential equation be wrong if I take the step too small?
Using too small of a timestep can lead to accuracy issues due to finite precision. Since you can use derivative approximations to derive time integration schemes, it is fair to look at how derivative ...
- 3,878
8
votes
Accepted
8
votes
ODE $x''(t)+\eta x'(t)+x(t)=0$ with the $\eta$ extremely small
The problem you have is what is called "stiff": it has two time scales, namely one for the oscillation (which is ${\cal O}(1)$) and one for the damping (which is ${\cal O}(\eta^{-1})$). If $\eta$ is ...
- 53.7k
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