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Questions tagged [ode]

Ordinary Differential Equations (ODEs) contain functions of only one independent variable, and one or more of their derivatives with respect to that variable. This tag is intended for questions on modeling phenomena with ODEs, solving ODEs, and other related aspects.

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17
votes
4answers
4k views

The definition of stiff ODE system

Consider an IVP for ODE system $y'=f(x,y)$, $y(x_0)=y_0$. Most commonly this problem is considered stiff when Jacobi matrix $\frac{\partial f}{\partial y}(x_0,y_0)$ has both eigenvalues with very ...
8
votes
2answers
4k views

Which Runge-Kutta method is more accurate: Dormand-Prince or Cash-Karp?

I simply want to know whether the Dormand-Prince Numerical Method or the Cash-Karp Numerical Method is more accurate.
2
votes
1answer
2k views

How can I implement the implicit Euler method for a small nonlinear system of ODEs?

I am trying to solve a system of coupled ODEs: $$ \begin{align} \frac{dn_A}{dt} & = e\left[j(t) - f\, θ_H\sinh\left(\frac{g\,n_A}{T}\right)\right] \\ \frac{dθ_H}{dt} & = a\left[bP\,(...
11
votes
3answers
1k views

Numerically stable explicit solution of small linear system

I have an inhomogeneous linear system $$ Ax=b $$ where $A$ is a real $n\times n$ matrix with $n\leq 4$. The nullspace of $A$ is guaranteed to be of zero dimension so the equation has a unique ...
9
votes
2answers
667 views

How to model a fishing rod (or a rope)?

I wish to model a fishing rod (or a rope) by joining short segments. (The segments may have equal (short) length but each segment should be assigned its own individual mass.) One segment will ...
1
vote
1answer
808 views

How can I call the Boost C++ odeint Runge-Kutta integrator for a system of ODEs?

I would like to use Boost C++ odeint Runge-Kutta integrator on a system that looks like this : $$\ddot x = - \frac A{||x||^3} * x $$ $ x $ is a vector in 3D space, so basicaly $ x(i, j, k) $ $ \...
39
votes
3answers
3k views

What's the state of the art in parallel ODE methods?

I'm currently looking into parallel methods for ODE integration. There is a lot of new and old literature out there describing a wide range of approaches, but I haven't found any recent surveys or ...
24
votes
2answers
2k 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 ...
6
votes
2answers
199 views

Why naively chopped finite difference matrix works for different ODE boundary conditions

We know finite difference method (FDM) can replace $y''(x)$ as $\frac{1}{h^2}[y(x+h)+y(x-h)-2y(x)]$ or so. One naive way to write down the matrix of the differential operator is like the following, ...
3
votes
1answer
434 views

Comparing Algorithmic complexity, ODE Solvers (Big O)

I am currently using the following three methods to solve differential equations: 4th order Runge Kutta Method Euler Method Internal scipy methods: ...
12
votes
1answer
215 views

Algorithms for linear system of ODEs

I wonder: what is the best algorithm to solve \begin{equation} \frac{du}{dt} = Au \end{equation} Where $A$ is a real $n\times n$ matrix. A is not explicitly time dependent, usually sparse but not ...
6
votes
2answers
2k views

Intermediate values (interpolation) after Runge-Kutta calculation

I have a numerical ODE simulation that I computed at fixed time step $h$ using a 4-th order Runge-Kutta method (RK4), producing a series of results $(x_1,y_1), (x_2,...
5
votes
2answers
6k views

How to implement Newton's method for solving the algebraic equations in the backward Euler method

Can you explain me how does the backward Euler method works? I have seen the formula and try to understand the method, but what I can't understand is why and how to use the Newton-Rapson method. Do ...
15
votes
3answers
1k views

Numerical methods for discontinuous r.s. ODEs

what are state of art methods for numerical solution of ODEs with discontinuous right side? I'm mostly interested piecewise-smooth right side functions, e.g. sign. I'm trying to solve the equation of ...
7
votes
2answers
2k views

How do I solve a boundary value ODE in MATLAB?

Specifically, ode15i. I have ode15i solving a system of 5 first order implicit odes in 5 variables with an initial condition (made consistent by decic). It's great for what I need, except I need to ...
7
votes
1answer
435 views

Why are functional representations of systems important in numerical applications?

I tried asking a similar question in SE.Physics, and I got some information regarding the abstract side of this, but I figured I should post here to get more complete information about the numerical ...
7
votes
1answer
291 views

Numerically solving systems of about 100 ODEs

I am looking to solve large systems of non-linear ODEs. There appears to be a very large list of methods available varying in complexity, and I have a hard time searching through them and picking one. ...
7
votes
1answer
312 views

Richardson extrapolation for strong rate of convergence of SDE

Is it possible to apply Richardson extrapolation with Euler-Maruyama scheme to improve strong rate of convergence of stochastic differential equations?
4
votes
1answer
2k views

Python: Grid with step control ODE solver

I have a problem in physics formulated via an ODE. Now I like to solve it numerically using Pythons scipy.integrate and the therein complex_ode. I figured out how and it works but now I like to ...
3
votes
1answer
121 views

Integration of differential equation with orthogonality constraint

Lets say I have a system of differential equations which has the form $$\dot{C}_{\alpha,\beta,m} = f_{\alpha,\beta,m}(C_{\alpha,\beta,1},\ldots,C_{\alpha,\beta,N};t).$$ The $f$s are some functions of ...
3
votes
1answer
1k views

Solving large, non-linear systems of ODEs numerically: what do I need to consider in order to figure out which solver to use?

I would prefer recommendations that don't require the use of proprietary tools (such as Matlab). I know of two ODE solving options for the Python ecosystem: PyDSTool (Dopri, Radau, other Runge-Kutta ...
2
votes
1answer
97 views

Formulation of the least-squares parameter estimation problem

I have a system of 10 ordinary differential equations of the form, $$\frac{dy_1}{dt} = f1(V1,k1,y1,y2)\\ \vdots \\ \frac{dy_{10}}{dt} = f_{10}(V_{10},k_{10},y_{9},y_{10}) $$ I want to estimate the ...
2
votes
1answer
119 views

Implementation details for high order IMEX methods by Kennedy and Carpenter

This question is a continuation of Fourth order IMEX Runge-Kutta method, concerning the implementation. Is seems to me that the first implicit stage value involves a direct evaluation, rather than ...
1
vote
0answers
2k views

How to solve an ode with stochastic time-dependent input

I am trying to repeat an example I found in a paper. I have to solve this ODE: $25 \ddot{x} + 15 \dot{x} + 330000 x = p(t)$ where $p(t)$ is a white noise sequence band-limited into the 10-25 Hz ...
0
votes
2answers
135 views

Non-linear Boundary Value Problem. How to compute the Jacobian?

Consider a Boundary Value Problem: $$ \delta u''+u(u'-1) =0 \Leftrightarrow u''=\frac{-u(u'-1)}{\delta}=:f(t,u',u), \\ u(0)=a, u(1)=b $$ $\delta,a,b$ are known parameters. I want to implement Newton'...
12
votes
3answers
3k views

Choice of step size using ODEs in matlab

Hey there and thanks for giving time to look at my question. This is a updated version of my question which I posted earlier in physics.stackexchange.com I'm currently studying a 2D exciton spinor ...
8
votes
1answer
557 views

Which numerical methods preserve time reversal symmetry?

If I have a physical system which contains a time reversal symmetry (for example a Hamiltonian $H(x,p)=p^2/2m + V(x)$ with $V(x)$ real) and I want to solve the differential equations which describe ...
8
votes
2answers
752 views

How to solve the stiff equation in this Restricted Three Body Problem numerically?

I've come across a stiff equation in solving the Circular Restricted Three Body Problem. [An object is moving considering the effect of the gravitational forces caused by two gravitational sources ...
5
votes
3answers
213 views

Is there a way to reduce aberration in computations of planets' trajectories?

I don't think the title is very accurate , sorry for that. I simulate bodies in space using two timestep: the TIMESTEP is the Δt wich I use to make the calculation and XTIME is the number of times ...
4
votes
1answer
1k views

How to find conditional Lyapunov exponents

For a synchronized ODE system,I wanted to know if there is a program code in MATLAB available for plotting the conditional Lyapunov exponent. For example, synchronization of identical Rossler system ...
4
votes
0answers
75 views

Large residual when integrating 2nd order ode close to singularity with SciPy ode / ODEPACK

I am trying to integrate a 2nd order ODE with a singularity at close to the initial condition. Why do I get large residuals when I plug-in the result of my integration back into the ODE? The equation ...
4
votes
3answers
2k views

4th-order Runge-Kutta method for coupled harmonic oscillator

I’m attempting to write a C program to gather values from a coupled spring system: There is a wall, connected to a mass $m_1$ by a spring, then this mass is connected to a second mass $m_2$ by another ...
3
votes
2answers
293 views

Nanoseconds vs. picoseconds in numerical quantum problems with Matlab ODEs

Hello there and thanks for taking a look at this problem. This problem is related to my previous question and I will therefore use a similar introduction from, Choice of step size using ODEs in ...
3
votes
2answers
218 views

Determine numerical infinity for Schrodinger equation $−\psi''(x) + x^ 2 \psi(x) = E\psi(x)$

Consider the following Schrodinger equation for the harmonic oscillator with real $x$: $$ −ψ''(x) + x^ 2 ψ(x) = Eψ(x). $$ I solve the last equation using shooting method and implicit Runge-Kutta ...
1
vote
1answer
2k views

2D cross section from 3D surface

I am trying to apply the "restoring force surface" method to a dynamic linear system. The idea behind this method is that, knowing acceleration, displacement, velocity and input force it is possible ...
0
votes
1answer
2k views

scipy.integrate.odeint: how can odeint access a parameter set that is evolving independently of it?

I might have some non-linear ODEs that are being solved by scipy.integrate.odeint. However, a parameter at each time step might have to be updated by using a non-DE ...
-1
votes
1answer
318 views

Which ODE solver in Matlab allows me to advance in just one timestep only

[t,y] = odeXX(odefun,tspan,y0) I have a solver odeXX, and the tspan = [0 0.0001]. It seems that for any ode solver in MATLAB, they integrate by breaking ...