# 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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### 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 ...
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### 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 ...
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### How does one test a numerical ODE solver implementation?

I'm about to start working on a software library of numerical ODE solvers, and I'm struggling with how to formulate tests for the solver implementations. My ambition is that the library, eventually, ...
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### How can the gravitational n-body problem be solved in parallel?

How can the gravitational n-body problem be solved numerically in parallel? Is precision-complexity tradeoff possible? How does precision influence the quality of the model?
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### Why are higher-order Runge–Kutta methods not used more often?

I was just curious as to why high-order (i.e. greater than 4) Runge–Kutta methods are almost never discussed/employed (at least to my knowledge). I understand it requires greater computational time ...
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### 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 ...
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### Options for solving ODE systems on GPUs?

I would like to farm out solving systems of ODEs onto GPUs, in a 'trivially parallelisable' setting. For example, doing a sensitivity analysis with 512 different parameter sets. Ideally I want to do ...
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### Why does the numerical solution of an ODE move away from an unstable equilibrium?

I wish to simulate the behaviour of a double-pendulum-like system. The system is a 2-degrees-of-freedom robot manipulator that is not actuated and will, therefore, behave mostly like a double-pendulum ...
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### 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 ...
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### Optimal ODE method for fixed number of RHS evaluations

In practice, the runtime of numerically solving an IVP $$\dot{x}(t) = f(t, x(t)) \quad \text{ for } t \in [t_0, t_1]$$ $$x(t_0) = x_0$$ is often dominated by the duration of evaluating the right-...
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### Best practices for describing agent-based models

I work fairly heavily in mathematical biology/epidemiology, where most of the modeling/computational science work is still dominated by sets of ODEs, admittedly sometimes fairly elaborate sets of them....
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### solving coupled ODEs with initial-value and final-value constraints

The essence of my question is the following: I have a system of two ODEs. One has an initial-value constraint and the other has a final-value constraint. This can be thought of as a single system with ...
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### 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 ...
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### Is there an open source set of ODE solvers for C that use the native C99 complex type?

I've been using GSL as the foundation of many of my simulations, but it's a little bit overkill for my purposes and it defines its own complex type for legacy reasons. Rather than code my own Runge-...
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### 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 ...
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### Runge-Kutta and Reusing Datapoints

I am trying to implement the fourth order Runge-Kutta method for solving a first order ODE in Python i.e. $\frac{dy}{dx} = f(x,y)$. I understand how the method works, but am trying to write an ...
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### 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 ...
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### How do you improve the accuracy of a finite difference method for finding the eigensystem of a singular linear ODE

I am attempting to solve an equation of the type: $\left( -\tfrac{\partial^2}{\partial x^2} - f\left(x\right) \right) \psi(x) = \lambda \psi(x)$ Where $f(x)$ has a simple pole at $0$, for the ...
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### Can I use an explicit time stepping scheme to determine numerically whether an ODE is stiff?

I have an ODE: $u'=-1000u+sin(t)$ $u(0)=-\frac{1}{1000001}$ I know that this particular ODE is stiff, analytically. I also know that if we use an explicit (forward) time stepping method (...
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### Dynamically ending ODE integration in SciPy

I have a light ray moving through space-time, i.e. a curve in $\mathbb{R}^4$, parametrized by some variable λ. The exact trajectory, i.e. the coordinate functions $x^μ(λ)$ of the curve are given by ...
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### 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.
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### Symbolic solution of a system of 7 nonlinear equations

I've got a system of ordinary differential equations - 7 equations, and ~30 parameters governing their behavior as part of a mathematical model of disease transmission. I'd like to find the steady ...
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### Reference request: Rigorous analysis of algorithms for PDE and ODE

I'm interested in suggestions for book references on the subject of numerical PDE and ODE, in particular, a rigorous analysis of such methods in a manner written for professional mathematicians. It ...
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### Constructing explicit Runge Kutta methods of order 9 and higher

Some older books I've seen say that the minimum number of stages of an explicit Runge-Kutta method of a specified order is unknown for orders $\geq 9$. Is this still true? What libraries are there ...
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### 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 ...
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### Numerics: How do I renormalize the following ODE

This question is more about how to tackle a problem numerically. In a small project I wanted to simulate the coorbital motion of Janus and Epimetheus. This is basically a three body problem. I ...
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### How to find Lyapunov exponent for coupled system

Answer gives a software for calculating conditional Lyapunov exponent (CLE) for coupled oscillators in chaos synchronization. However, it is hard to follow and there is no graphical output of the ...
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### ODEs vs DAE vs ADE?

I am totally confused between ODEs which I am familiar with, and differential algebraic equations (DAE) and Algebraic Differential Equations (ADE). Are they the same but just different names or what ...
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### Is the shooting method the only general numerical method for solving nonlinear boundary value ODEs?

During my wandering in Mathematica.se, I gradually noticed that a certain kind of differential equation solving problem is "troubling" us all the time, that is, the boundary value problem (BVP) of ...
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### How can I numerically solve an ODE to $N$ provably correct digits?

Suppose we have an initial value problem of the form $$\frac{\mathrm{d} \mathbf{x}}{\mathrm{d} t} = f(\mathbf{x}) \qquad \mathbf{x}(0) = \mathbf{x}_0$$ where $\mathbf{x}_0 \in \mathbb{R}^n$ is known ...
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