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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9
votes
2answers
6k 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
86 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
1k 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
95 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 ...
1
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1answer
136 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
votes
1answer
174 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
votes
2answers
149 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
vote
0answers
933 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: ...
5
votes
0answers
338 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: $$ \frac{1}{x^2}\frac{d}{dx}\left(Gam \frac{dL}{dx}\right)+L\left(\frac{a^2x^2}{Gam}-m^2\right)=0 $$ ...
0
votes
1answer
165 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
551 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
534 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 ...
2
votes
0answers
89 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 ...
3
votes
0answers
109 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
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2answers
74 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
224 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
558 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
156 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 ...
5
votes
2answers
208 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
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1answer
287 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
2k 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
2answers
175 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
242 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
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0answers
117 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
892 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
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0answers
814 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
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1answer
155 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
405 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
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2answers
154 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 ...
5
votes
2answers
4k 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
943 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
67 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
4k 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
56 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
vote
0answers
55 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
64 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 ...
0
votes
1answer
149 views

ODE System doesn't work when step size (h) is bigger than 1

I am a beginner to Python. Currently I'm writing a code for developing a simple solver for non-linear ODE systems with initial value. The equations of the system are The function of myu is evaluated ...
2
votes
1answer
1k views

Set of linear ordinary differential equations with a mass matrix

What methods are known for efficiently solving a large set of linear homogeneous ordinary differential equations of the following form? \begin{equation} \mathbf{B} \frac{d\mathbf{y}}{dt} = \mathbf{A} \...
2
votes
0answers
2k views

Using RK2 Method to solve the simple harmonic oscillator of a horizontal mass on a spring (1D)

Being new to numerical analysis techniques, in particular RK2, I decided the best way to jump in is by using python to solve the well known mass-spring oscillator using RK2 techniques. My problem is ...
1
vote
1answer
145 views

Fast table interpolation on regular time data for ODE

I am using scipy.integrate.odeint to simulate the reaction of a system with known input signals via integration. The simplified code below illustrates what I'm ...
0
votes
1answer
121 views

Stability of dark solitons in a harmonic trap

This question is based upon a research article which I am trying to reproduce. One of the main result of this paper is the condition on transverse confinement of the Bose-Einstein Condensate(BEC) to ...
1
vote
1answer
142 views

VTK: Missing Streamlines due to error in Runge-Kutta method?

We are using Kitware VTK to visualize our models. When we display streamlines, it sometimes happens that a single streamline is left out. This can be especially seen when duplicating a model several ...
4
votes
0answers
538 views

Convergence rate of Picard iterations

Given a first order ODE $y'(x)=f(x,y)$ with the initial condition $y(x_0)=x_0$ such that it satisfies Picard thoerem of existence and uniquness, one can compute the solution by Picard iterations : $$ ...
1
vote
1answer
279 views

Trouble getting steady-state solution by solving system of nonlinear algebraic equations in MATLAB

Background I have a stiff system of 6 ODEs, represented in MATLAB as follows: ...
2
votes
1answer
113 views

Solving a system of DAEs versus ODEs (which is preferable)

Assume I have the following system of equations. I am trying to figure out if it's better for me to be solving this as a system of ODEs or as a system of DAEs. The real code can have up to a dozen or ...
1
vote
0answers
203 views

How to implement chaotic sender and receiver with ordinary differential equations?

I am referring this paper, and trying to implement chaotic sender and receiver, to decode message, as given in section $V^{th}$ Chaotic signal masking. The process that I want to implement is figure ...
1
vote
2answers
109 views

Trotter expansions in ode solver?

Trotter expansions say: $$ e^{A+B} = \lim_{P\to\infty} \big(e^{A/2P} e^{B/P} e^{A/2P} \big)^P. $$ With $P = 2$, it becomes (with high accuracy) $$ e^{A/4} e^{B/2} e^{A/2} e^{B/2} e^{A/4}. $$ Let's ...

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