All Questions
26 questions
51
votes
2
answers
14k
views
What does "symplectic" mean in reference to numerical integrators, and does SciPy's odeint use them?
In this comment I wrote:
...default SciPy integrator, which I'm assuming only uses symplectic methods.
in which I am refering to SciPy's odeint, which uses ...
- 1,088
1
vote
1
answer
15k
views
Python Finite Difference Schemes for 1D Heat Equation: How to express for loop using numpy expression
Hello all,
I've recently been introduced to Python and Numpy, and am still a beginner in applying it for numerical methods. I've been performing simple 1D diffusion computations. I suppose my ...
13
votes
2
answers
8k
views
Numerical computation of Lyapunov exponent
I'm trying to compute the Lyapunov exponent for a smooth continuous time dynamical system(say, $\dot{\bar{x}} = f(\bar x)$). I using the QR decomposition method. Here are the steps that I follow.
...
- 263
10
votes
3
answers
2k
views
Numerical integration of ODEs: Why does higher accuracy and precision not lead to convergence?
I'm trying to integrate numerically a non-linear ODE that, on paper, is simple (just the damped, forced pendulum), but that, as well known, its dynamics displays a chaotic behavior and is very ...
- 233
7
votes
2
answers
16k
views
Solving coupled differential equations in Python, 2nd order
I have a system of coupled differential equations, one of which is second-order. I am looking for a way to solve them in Python. I would be extremely grateful for any advice on how can I do that!
$k$...
- 183
6
votes
3
answers
727
views
Finite difference for 1D wave equation: why the spike initial data results in a noisy output?
I am using a second-order finite difference in space and time approximation for the 1D wave equation.
No source but initial data: $I(x)=\mathrm{e}^{-400 (x-0.5)^2}$.
Velocity $c=1$, $nx=501$, $nt=...
- 279
4
votes
1
answer
534
views
Fast and free server for computing
I have to calculate a huge differential equation. With my laptop, it's going to be computed for several days.
Is there a free (I need just for 3 days) fast server for scientific calculations?
My ...
- 143
4
votes
3
answers
1k
views
How can I numerically integrate the Kepler problem?
I tried to solve a simple Kepler problem numerically.
I have discrete time steps, a starting position $(x_0,y_0)$ and starting velocity $(u_0, v_0)$.
I used this iteration by calculating the forces ...
- 151
3
votes
1
answer
1k
views
Boundary value problem with singularity and boundary condition at infinity
I'm trying to solve the following boundary value problem on $[0,\infty]$:
$$f^{\prime \prime}=-\frac{1}{r} f^{\prime}+\frac{1}{r^{2}} f+m^{2} f+2 \lambda f^{3}$$
$$f(0)=0 \ ; f(\infty)=\sqrt{-m^2/(2\...
- 57
2
votes
1
answer
417
views
MATLAB's ode45 not dealing with initial conditions well [RESOLVED]
*Concern highlighted in yellow
*Solution at bottom
I have a differential equation to solve for the motion of an electron:
$$
\frac{d^2v}{dt^2} = \frac{1}{\gamma^6}\left( \frac{eE}{\tau m} - \left( \...
- 151
1
vote
2
answers
1k
views
How to set up the differential equation system to speed up computation?
I've set up a system of differential equations, obtained after discretizing pde, in the following way
...
- 433
0
votes
2
answers
503
views
Can the Crank-Nicolson Method Be used to Solve The Schrodinger Equation with a Time Varying Potential?
I have been following an excellent article about how to use the Crank-Nicolson method to solve the Schrodinger equation. In the article, it starts with a $V(x, y, t)$ but the potential seems to become ...
- 33
5
votes
2
answers
3k
views
CUDA & Python for numerical integration and solving differential equations
Can anyone please suggest some libraries which allow use CUDA in Python for numerical integration and/or solving of differential equations?
My goal is to solve large (~1000 equations) of coupled non-...
- 181
5
votes
2
answers
775
views
Specifying ode solver options to speed up compute time
I'm specifying the 'JPattern', sparsity_pattern in the ode options to speed up the compute time of my actual system. I am sharing a sample code below to show how I ...
- 433
3
votes
1
answer
2k
views
Solving the heat diffusion equation with source term
I am trying to solve the 1-D heat equation numerically with a variable source term. The system is basically a tank containing styrene in which it polymerizes to liberate heat. I have assumed that the ...
3
votes
2
answers
928
views
Python bifurcation diagram of seasonally forced epidemiological models
TL:DR
How can one implement a bifurcation diagram of a seasonally forced epidemiological model such as SEIR (susceptible, exposed, infected, recovered) in Python? I already know how to implement the ...
- 265
3
votes
2
answers
2k
views
Solving numerically the 1D Kuramoto-Sivashinsky Equation using spectral methods
Lately, I've been trying to solve numerically the 1D Kuramoto-Sivashinsky Equation using spectral methods.
Let $\nu$ be the viscosity and $[0,L]$ the domain. The 1D equation is,
$$
u_t + uu_x + u_{xx} ...
- 173
3
votes
2
answers
523
views
Runge Kutta and Milstein – system of second-order coupled differential equations with noise
I would like to solve a system of second-order differential equations to describe the dynamics of a system of particles.
Two Newton-like forces are responsible for the motion of each particle $i$: A ...
- 31
2
votes
1
answer
411
views
Where am I making a mistake in solving the heat equation using the spectral method (Chebyshev's differentiation matrix)?
I would like to numerically solve the following heat equation problem:
$$ u_t = \Bigg(2{a \over l}\Bigg)^2 u_{xx} \tag 1$$
$$ x \in [ -1, 1 ] \tag 2$$
$$ u(x, 0) = 0 \tag 3$$
$$ u(1, t) = A \sin \Bigg(...
2
votes
1
answer
424
views
Numerical computation of Lyapunov exponents: how to find convergence or non-convergence efficiently?
I am wondering what are the standards for convergence of Lyapunov exponents (and Kaplan-Yorke dimension)? For example, I have a MATLAB code to calculate Lyapunov exponents for the classic Lorenz ...
- 307
2
votes
3
answers
195
views
Numerically finding constants of motion
Given a set of ODE's $ \dot{z} = f(z) $ (or discrete time $ z_{t+1} = f(z_t) $), is there a way to numerically find constants of motion?
For $ f(z_t) \approx M z_t $, diagonalizing the matrix $ M $ ...
user23933
1
vote
1
answer
2k
views
How to solve heat equation in spherical coordinates with finite differences?
I have a problem dealing with heat transfer which is spherically symmetrical. I was thinking it should be possible to solve this as a 1d problem in spherical coordinates using the radius only.
...
- 111
1
vote
1
answer
2k
views
scipy odeint: excess work done on this call and very sensitive to initial value
I am trying out odeint and received the error
'Excess work done on this call (perhaps wrong Dfun type).'.
The values returned are also super sensitive to small ...
- 13
1
vote
1
answer
349
views
Crank-Nicholson for diffusion-advection vs diffusion equation
Let's consider the following 1D diffusion equation:
$\frac{\partial u}{\partial t} = xk \frac{\partial}{\partial x}(\frac{1}{x}\frac{\partial u}{\partial x})$
where we assume that the diffusion ...
- 172
1
vote
2
answers
194
views
Solving a parameter estimation problem using trajectory optimization
This is a follow-up to my previous question here
I've the following system of equations for studying information flow in the below graph,
$$ \frac{d \phi}{dt} = -M^TDM\phi + \text{noise ...
- 433
0
votes
2
answers
647
views
Why is this scipy.root code not converging?
I'm running a test problem to set up larger problems. Solving the simple unsteady heat equation via finite differences:
$$ \frac{\partial T}{\partial t} = \alpha \frac{\partial^2T}{\partial x^2}$$
$\...
- 133