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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3
votes
1answer
95 views

Ordinary differenial equation with numerical right hand side

I'm interested in solving a differential equation in which I don't know the analytical form of the right hand side, I only know its numerical value for a finite set of values of the independent ...
0
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1answer
135 views

How do I find the default ODE solver tolerances in Matlab?

You can set the Absolute or Relative ODE solver tolerances in Matlab with the options structure from an odeset command. But how do I find the default values for the ...
4
votes
2answers
129 views

Estimating error at mid timesteps for Runge-Kutta methods

When a timestep $h$ is rejected using a Runge-Kutta pair, such as Dormand–Prince, the algorithm resumes from the same initial point $t$ with a smaller timestep. A different idea is to resume at an ...
9
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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
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0answers
84 views

Solving an ODE from many starting points

I have a situation where I'm interested in solving the same ODE many times, from different initial locations. More precisely, I'm interested in solving an ODE of the form \begin{align} \frac{dx}{dt} &...
5
votes
1answer
650 views

Solving for a set of coupled ODEs to get correct variable values

My question is about how I can solve a coupled system of ODE's, and print out the variables in a plot. I am solving for an q value and an e value, seen in this set of coupled ODE's below: $$ \begin{...
3
votes
2answers
93 views

Error control and sequence acceleration at the same time

In a posteriori error control for solving ODEs, one typically computes two different approximate solutions, one of which being "more accurate" and one of which being "less accurate". If $y_q^{n+1}$ is ...
39
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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 ...
1
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1answer
102 views

Numerical solution for eigenvectors and eigenvalues of a Sturm-Liouville problem

I have to deal with the following problem in my research: $$\left[\frac{1}{D}F_{x}\right]_{x} + \frac{f D_{x}}{c D^{2}}F = 0$$ with boundary conditions $$F(0) = 0$$ $$F_{x}(L) = 0$$ where $f$ is ...
2
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1answer
137 views

Adaptive Timestepping for Stong Stability Preserving (SSP) Runge-Kutta Methods

Are there error estimators and research on adaptive timestepping schemes for SSPRK methods? My Googling could not uncover papers which addressed this, so I was wondering if there was anything ...
5
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2answers
134 views

Methods for precise solution of an ODE whose solution terminates at a singularity

I'm working on a fun open-source project to calculate the trajectories of objects near black holes. This is obviously not the first time anyone has done this sort of thing, but I have some design ...
1
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0answers
597 views

Which solvers for BVP in python are the best? Is there something better that scipy.integrate.solve_bvp?

I am trying to solve a boundary value problem with Python. I have been using scipy.integrate.solve_bvp but the result that it is giving me is completely wrong. Basically my code is as follows: ...
3
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0answers
212 views

Avoiding divergent solutions with `odeint`? shooting method

I am trying to solve an equation in Python. Basically what I want to do is to solve the equation: $$ \frac{1}{x^2}\frac{d}{dx}\left(Gam \frac{dL}{dx}\right)+L\left(\frac{a^2x^2}{Gam}-m^2\right)=0 $$ ...
1
vote
1answer
107 views

Parallel integration of dynamical systems

I need to solve the following problem: $$ \begin{cases} \dot{\vec{x(t)}} = A\vec{x(t)} + u(t)D\vec{x(t)} + u(t)\vec{b}, & x \in (0, T), \\ \vec{x(0)} = \vec{0}, \end{cases}$$ where $u(t)$ is known ...
0
votes
1answer
141 views

maltab ode solver- user defined criteria to stop calculations

is there a way to add a user defined convergence criteria to an ode solver so that the solution is stopped? I know that Matlab uses absolute and relative tolerances but would that suffice in solving ...
3
votes
2answers
267 views

ODE system with discontinuous right-hand-side

I have a 1st order ODE system. One of the equations is piecewise function of 2 of the dependent variables. I try to solve it in Python environment. \begin{align}\dot x_1 &= x_2 - x_3\\ \dot x_2 &...
1
vote
1answer
168 views

System of ordinary differential equations - time complexity of initial value problem

I am interested in knowing what the time complexity is (in Big-$\mathcal O$ notation) for solving system of $N$ differential equations? I am using ode15s in ...
0
votes
1answer
536 views

How to implement an integral condition when solving a BVP in MatLab

I am trying to solve a coupled system of ODE's using MATLAB's bvp4c function. I want to impose the condition that $$\int_{0}^{\pi} y_{1}(t) y_{1}(t) dt = 1,$$ where ...
2
votes
0answers
64 views

Finite differenced eigenvalue prob. of inhomogeneous boundary conditions?

I am basically asking about eigenvalue problems of differential equations using some finite difference method (FDM). Usually the system is subject to some boundary conditions (BC), e.g., Dirichlet or ...
2
votes
1answer
125 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 ...
3
votes
0answers
100 views

How to account for the interface between two different phases in a discretized diffusion model?

I have tried to set up a model for the diffusion of a gas into a liquid. The two media are next to each other and the geometry is spherical because the system should simulate the diffusion out of a ...
0
votes
1answer
104 views

Runge-Kutta timestep in atomic units

I'm using 4th order RK to solve the schroedinger equation in atomic units. Say I want to simulate 400fs in intervals of h=10fs, then in atomic units this is h=413a.u and 400fs=16500a.u. 4RK involves ...
0
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2answers
71 views

Finite difference for 2nd order ode $y'^2+y y''+\frac{2}{x} y y' -0.1 y^2=0$ with $y'(1)=0$ and $y(1)=1$

How to solve second order non-linear ODE $$y'^2+y y''+\frac{2}{x} y y' -0.1 y^2=0$$ subject to $y'(1)=0$ and $y(1)=1$ over the interval $0 < x \le 1$. I turned the equation to a PDE $y'^2+y y''+\...
6
votes
2answers
201 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, ...
5
votes
2answers
349 views

Do there exist low-storage Runge–Kutta methods with an order larger than four?

I’m trying to find a Runge–Kutta integrator that only requires the absolute minimum of storage, i.e. it fulfills one of the following three, presumably equivalent criteria: Each evaluation of the ...
0
votes
1answer
113 views

Is the time step size of a Rosenbrock method for stiff systems iteratively calculated?

I have an ODE system of the general form y' = k(y)(x) + q(z)(x) x' = a(z)(x) + b(x)(x) where k,q,a and b are also dependent on the states x and y. The ...
3
votes
1answer
560 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: ...
10
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0answers
219 views

Inverse problem in linear ODE

I have a linear ordinary differential equation (ODE) with a system matrix with constant coefficients: $$\dot{y}(t) = \mathcal{A}\; y(t), \quad y(0) = y_0$$ with $y(t) \in \mathbb{R}^{n \times 1}$ and $...
5
votes
2answers
159 views

Algorithm for finding initial conditions of differential equations given trajectory

Let's say I'm given a system of three first-order differential equations in three variables, where all of the equations are known, and we additionally know the trajectory of two of the variables at a ...
1
vote
1answer
188 views

Bulirsch-Stoer algorithm to solve simple chemistry. Spanning long time intervals after stationarity

Hi all and thank you in advance. I am working on a time-dependent transport-chemistry model to study the composition of planetary atmospheres. The equations are the following $$\frac{\partial n(z,t)}...
0
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1answer
958 views

4th order runge-kutta and harmonic oscillator [closed]

I am trying to solve equations of motion for an harmonic oscillator using 4th order runge kutta method, but as a result I get almost constant velocity and position; I feel that the problem is that I ...
1
vote
1answer
186 views

Method to solve linear, first order ODE of generalized matrix matrix form

The equation and its meaning: Consider two sets $(A)_{l=0,...,m_a},$,$(B)_{l=0,...,m_b}$ of hermitian matrices and a set of positive semidefinite matrices $(C)_{l=0,...,m_c}$. Each matrix has the ...
1
vote
2answers
146 views

Learning differential equations: a textbook

I am a Computational Biology master's student and I would like to study differential equations (both ODEs and PDEs). I tried to have a look at the available textbooks but what I found is either too ...
3
votes
2answers
203 views

Failing integration with the radau5-implementation in DotNumerics, over a discontinuity

Summary We are trying to solve some models of emission of substances to the air. In these models, emission stops at a certain point. We are using the DotNumerics implementation of radau5 We had ...
1
vote
0answers
111 views

Stiffness emerges as number of ODEs increases

I want to solve a system of ordinary differential equations with Matlab. I need this to solve a mechanical engineering related problem. If $n$ is the number of degrees of freedom of my mechanical ...
0
votes
1answer
624 views

Speeding up ODEINT stiff example [closed]

I am trying the implement the following odeint solver example but my differential equations are different. https://github.com/headmyshoulder/odeint-v2/blob/master/examples/stiff_system.cpp The ...
2
votes
1answer
44 views

Verifying that ODE integration generates Theoretical Stationary distribution

I am trying to simulate an ODE, like $ \dot{x} = \xi(x) $ that should have a stationary distribution (a la Stat Mech). Assuming that my ODE algorithm generates time samples of my system state $ x $ ...
1
vote
0answers
491 views

One Dimensional Schrodinger's Equation solution using Numerov Method

I have been trying to solve Time Independent Schrodinger's equation in one dimension using Numerov Method as discussed in this excellent lecture notes I found on net. The Numerov method can solve an ...
0
votes
1answer
131 views

Parameter identification for first order ODE

I have two arrays $f(z)$ and $z$ both indexed by k and I want to solve $\frac{df}{dz}=\mu(1-f)$ to find $\mu(z)$ What would be the best numerical method to solve this equation?
0
votes
1answer
289 views

``scipy.odeint`` giving different answer than analytical

I was using scipy.integrate.odeint function , the ode is $$\frac{y\ dx - x\ dy}{(x+y)^2} + dy = dx$$ with solution $$y^2 - x^2 - y = c (x + y)\ .$$ Solving it ...
0
votes
2answers
133 views

implicit odes solution using fdm

I am solving a non linear second order implicit initial value problem using finite difference method, but my results do not converge. Please guide me with an example, how we can apply finite ...
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 ...
4
votes
2answers
3k views

BDF2 and TR-BDF2: what is better?

What method of numerical solving ODEs is better? BDF2 or TR-BDF2? Namely, what advantages has TR-BDF2 over BDF2? The BDF2 method requires the values of $y_{n-1}$ and $y_n$ for computing $y_{n+1}$ ...
1
vote
1answer
833 views

How can I solve coupled equations by the method of line(MOL)?

I want to solve 3 coupled PDEs equations. They depend on time, radius and length. I used the method of lines (MOL) and converted them to a system of ODEs in time. Now I want to solve them using MATLAB....
1
vote
1answer
65 views

Calculation of integration curve

Consider the unit circle $x^2 + y^2 = 1$, it also represents the solution to the ODE $$ x + y \frac{dy}{dx} = 0; y(0) = 1 $$ Suppose we don't know how to solve the above equation analytically (there ...
-1
votes
1answer
2k views

Using scipy.odeint to solve coupled equations [closed]

I have a set of three coupled autonomous equations: ${y_{1}}\prime = y_{1}(\frac{\Omega_{m}}{y_{1}^3} + \frac{y_{3}^2}{6.0} + \frac{V(y_{2})}{2.H_{0}^2})$ $y_{2}\prime = y_{3}$ $y_{3}\prime = -3\frac{...
6
votes
3answers
1k views

4th order Runge-Kutta for $y' = y$

My question is quite simple, but the more I look at it, the less content I am. My question is how to do a RK4 method for $y'=y$. At first I would assume the following: $$k_1=y_n$$ $$k_2=y_n+\frac{...
0
votes
1answer
51 views

Can RK5 be considered as part of Predictor–corrector method?

It seems that the 5th order is the corrector and the 4th order is the predictor. Can RK5 be considered as part of Predictor–corrector method?
1
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0answers
46 views

Solve ODE with two unknown functions

I want to solve a diffusion-convection problem numerically. I start from a PDE of the form $$\partial_t f(x,t) = \partial_x(K(x)\;f(x,t))+\partial_{xx}\;f(x,t)^{m}\;.$$ It is possible to calculate a ...
0
votes
1answer
59 views

Examples of finding eigenfunctions of coupled DEs

I am looking for examples of numerically finding eigenvalues/eigenfunctions of coupled DEs. If anyone is able to point me towards any examples, preferably with code included, it would be much ...

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