41 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 ...
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35 votes
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 ...
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22 votes
Accepted

Puzzling remark about stability region of fifth-order Runge-Kutta method

van der Houwen's statement is correct, but it is not a statement about all fifth-order Runge-Kutta methods. The "Taylor polynomials" he is referring to are (as you seem to know) just the polynomials ...
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19 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 ...
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18 votes
Accepted

Applying the Runge-Kutta method to second order ODEs

There seems to be quite a bit of confusion about how to apply multi-step (e.g. Runge-Kutta) methods to 2nd or higher order ODEs or systems of ODEs. The process is very simple once you understand it, ...
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17 votes

Looking for Runge-Kutta 8th order in C/C++

If you're doing celestial mechanics over long time scales, using a classical Runge-Kutta integrator will not preserve energy. In that case, using a symplectic integrator would probably be better. ...
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17 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 ...
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16 votes
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 ...
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15 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 ...
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  • 1,839
15 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 ...
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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 ...
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12 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 ...
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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 ...
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  • 1,097
10 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 ...
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  • 216
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 ...
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9 votes
Accepted

Looking for Runge-Kutta 8th order in C/C++

Both GNU Scientific Library (GSL) (C) and Boost Odeint (C++) feature 8th order Runge-Kutta methods. Both are opensource, and under linux and mac they should be directly available from the package ...
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  • 2,473
9 votes
Accepted

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

There are many kinds of RK methods which 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 ...
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8 votes
Accepted

Stability analysis of Heun's method

Notice that $\hat C_*=(1-r F(h))\hat C_n$, but the sign in front of $r$ is lost when you use $\hat C_*$ inside $\hat C_{n+1}$. I'm looking at p.149 of Numerical Solution of Time-Dependent Advection-...
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  • 11.4k
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 ...
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7 votes
Accepted

What is the case of trade-off in different Runge-Kutta methods

There are so many Runge Kutta methods, including Dormand-Prince 45 Cash-Karp 54 Fehlberge (sic) 78 Is there any comparison between them? Well, sure. Here are some traits to ...
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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-...
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7 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 ...
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  • 6,066
7 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} &= ...
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  • 1,198
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: ...
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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 ...
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6 votes
Accepted

Solution blows up when using Runge-Kutta to solve simultaneous ODEs for liquid film equations

Bearing in mind that I have just a vague idea of what results you are expecting, I want to point out that I do not get the same ODE system as you do. I'm using Mathematica. Using your equations for $...
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  • 11.4k
6 votes

Solution blows up when using Runge-Kutta to solve simultaneous ODEs for liquid film equations

The first thing I can think of with your derivation is the $\mathbf{A}$ term. If at any point that term fails to be invertible, then your derivation is no longer valid. You should see the the value of ...
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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 ...
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6 votes
Accepted

Integration of differential equation with orthogonality constraint

In general, one cannot expect rk4 to maintain quadratic invariants of the system, it simply doesn't do that. Methods that do maintain specific invariants have to be specially devised — this usually ...
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  • 11.4k
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–...
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