# Tag Info

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### 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 ...

### 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 ...
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### CUDA & Python for numerical integration and solving differential equations

Julia's DifferentialEquations.jl is all GPU-compatible. If you make your arrays GPU-based arrays, then the solver recompiles to be all on the GPU (no data transfers). For example: ...
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### 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 ...

### How to solve a second order differential equation (diffusion) with boundary conditions using Python

I have found that I must keep the value of dt near dx or the results become unstable This behavior you have noticed is known as the Courant–Friedrichs–Lewy (CFL) condition. Indeed, there are ...
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It is not the Jacobian of $f$ that you need to use to propagate the local basis, but the Jacobian of the propagator of the numerical method. That is, if $$v_1=v_0+\Phi_f(v_0,dt)dt$$ then $$U_1=(I+... 8 votes Accepted ### Visualizing the solutions of the Differential equations by varying different parameters You can try Geogebra (it is free). With SolveODE command and sliders you can do what yo want. For the usage of SolveODE command see. For example by using following command ... 8 votes ### Finite difference for 1D wave equation: why the spike initial data results in a noisy output? I think you're probably seeing artifacts that are due to numerical dispersion. In brief, in the discrete case different (spatial) frequencies of a wave function will propagate at different phase/group ... 7 votes Accepted ### What is the case of trade-off in different Runge-Kutta methods There are so many Runge Kutta methods, including Dormand-Prince 45 Cash-Karp 54 Fehlberge (sic) 78 Is there any comparison between them? Well, sure. Here are some traits to ... 7 votes ### Specifying ode solver options to speed up compute time Julia's DifferentialEquations.jl has a lot of tooling for automatically deriving (sparse) matrices. For more information, see the JuliaCon 2020 video on Auto-Optimization and Parallelism in ... 6 votes ### Some questions on Trace (operators) on the boundary in the context of PDEs I think the quote from Wikipedia is misleading (and I should put that on my list of things to fix :-) ). You can think of the trace of a function u: \Omega \rightarrow {\mathbb R} that lives on a ... 6 votes ### Solving coupled differential equations in Python, 2nd order The first step is to transform the second order equation to a set of two coupled first order equations. Define an auxiliary function u(T) = \frac{dr(T)}{dT}. This results in the system$$\begin{...
Runga-Kutta schemes are multistage recipes for numerical discretizations of temporal derivatives. That is they tell to how to solve equations in the form $\dot{\mathbf{y}} = \mathbf{f}(\mathbf(y,t)$. ...