# Tag Info

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'...
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 ...
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 ...

### 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 ...
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 ...
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 ...
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 ...

### 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 ...

### 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 ...

### 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 ...

### 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 ...
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 ...
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} &= ...
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 ...
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 ...