44
votes
Accepted
BDF vs implicit Runge Kutta time stepping
Yes, there aren't too many resources on this for some reason. For a very long time, the standard goto was "just use BDF methods". This mantra was set in stone for few historical reasons: for ...
- 12.5k
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 ...
- 16.9k
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 ...
- 12.5k
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 ...
- 12.5k
16
votes
Accepted
Why is leapfrog integration symplectic and RK4 not, if the latter is more accurate?
TL;DR: It depends on what kind of accuracy you need.
Energy conservation does not automatically equal accuracy. Suppose, you want to simulate the solar system, and you are using a solver that – to ...
- 2,147
14
votes
Accepted
Easily understandable argument that normal Runge–Kutta methods cannot be generalised to SDEs?
Let's take a stochastic differential equation:
$$ X_t = f(t,X_t)dt + g(t,X_t)dW_t $$
Here's a few different arguments which lead to intuitive understandings of why the mathematics behind the higher ...
- 12.5k
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 ...
- 19k
13
votes
Energy conservation in RK4 integration scheme in C++
RK4 is not symplectic so it has no guarantee of energy conservation. Especially when solving an N-body problem where two bodies pass by close to each other the energy conservation can be violated ...
- 2,859
12
votes
Runge-Kutta in the presence of an attractor
The problem you are encountering is likely not a consequence of your choice of algorithm, but in fact a consequence of the resulting dynamical system after applying time reversal. Per the definition ...
- 3,101
12
votes
10th-order Runge-Kutta Method
Julia's DifferentialEquations.jl package (linked here via the subpackage OrdinaryDiffEq.jl which contains the source code of many of the methods) includes an implementation of a 10th order explicit RK ...
- 3,101
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 ...
- 3,293
10
votes
10th-order Runge-Kutta Method
You seem to be misunderstanding. Since you're referring to an explicit Runge-Kutta method, applying the method does not require that you solve any equations.
In Ernst Hairer's article, he is deriving ...
- 16.9k
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} &= ...
- 1,273
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 ...
- 12.5k
9
votes
What is the advantage of using a particular RK Scheme?
You're looking only at the errors themselves and not other properties of the solution.
There are sometimes good cases to consider lower-order schemes because they better preserve important ...
- 10.5k
8
votes
Accepted
How well do explicit Runge-Kutta "tableau" methods compare to the state of the art ODE solvers and when do they fail?
I think the confusion here is what exactly is a "tableau method" since it's used in our docs in a very specific way. A "tableau method" in DifferentialEquations.jl parlance is a method which is ...
- 12.5k
8
votes
Accepted
Is there a database/website with Butcher tableaus?
The three best sources of Butcher tableaus are, according to me, the reference books
Numerical Methods for Ordinary Differential Equations, 3rd ed., J.C. Butcher,
Solving Ordinary Differential ...
- 6,199
8
votes
Accepted
Motion of the particle trapped in potential
Your time step is far too large. Just looking at the first stage, you have $\dot{p}\approx10^{-12}$, so for the next stage, you will have something like $\dot{q} \approx h\dot{p}/m \approx 10^{-1}(10^{...
- 3,101
8
votes
Accepted
What is the advantage of using a particular RK Scheme?
There are lots of different properties which can be found in different time stepping schemes of the same order of accuracy:
Different stability properties. While it may not appear that way with the ...
- 2,859
7
votes
Do there exist low-storage Runge–Kutta methods with an order larger than four?
Another good starting point is the excellent paper of Kennedy, Carpenter, & Lewis (KCL2000):
https://doi.org/10.1016/S0168-9274(99)00141-5
My own paper focuses more on the mechanics of low-...
- 16.9k
7
votes
Accepted
The Formula of Explicit Runge-Kutta Fourteen order
The 14th order methods due to Feagin can be found in DifferentialEquations.jl. An example using them with 128-bit floating point arithmetic is as follows:
...
- 12.5k
7
votes
Accepted
Solving ODEs with nonlinear constraints
As @WolfgangBangerth already commented, this is usually referred to as a differential algebraic equation (DAE). These have their own challenges and there are special numerical methods for them.
In ...
- 16.9k
7
votes
Energy conservation in RK4 integration scheme in C++
Main error
As I pointed out in the previous questions of this series
RK4 integration of the three-bodies problem with C++
the primary problem is that the methods are not implemented correctly. The ...
- 6,159
7
votes
Accepted
Does this second-order implicit Runge-Kutta method have a name?
As Wolfgang stated in the comments, this is not a traditional RK due to the inconsistent time evaluations within a stage. At first it would seem it can't even be cast as an additive RK since terms ...
- 1,124
6
votes
Constructing explicit Runge Kutta methods of order 9 and higher
@DavidKetcheson's answer hits the big points: you can always construct methods of high enough order using extrapolation, that's a very pessimistic bound and you can always do a whole lot better, all ...
- 12.5k
6
votes
Looking for Runge-Kutta 8th order in C/C++
summarizing some points:
If it's a long-term integration of a non-dissapative model, a symplectic integrator is what you're looking for.
Otherwise, since it's an equation of motion, Runge-Kutta ...
- 12.5k
6
votes
Accepted
Type of Rosenbrock method by its coefficients
A search for the specific coefficients listed led me to the method ROS3PRL from
J. Sieber, Konvergenzanalyse und Numerische Tests für die Prothero–Robinson–...
- 1,124
5
votes
Accepted
Developping PDE with Python symbolically and numericaly
But how do you numerically (Runge-Kutta or whatever) solve the objects coming out of SymPy?
That depends a lot on your specific problem and what level of optimization you need.
Most ODE solvers (...
- 2,147
5
votes
Runge Kutta solution blows up for a first order ODE with very large coefficients
Though the following will not necessarily solve your problem, it is highly recommended to implement the ode dimensionless such that all relevant quantities are around 1 to stabilize solvers. The ...
- 1,335
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