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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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18 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'...
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0answers
41 views

Matlab simulation: conflicting results

I have implemented a simulation of the model (for $i=1,\ldots,N$ representing nodes of a graph) $$\frac{d\omega_i}{dt} = P_i -\alpha_i \omega_i + K \underset{i \neq j}{\sum_{i=1}^N} \sin(\phi_i - \...
2
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1answer
86 views

What does the exponential function mean in numerical ode solving formulas?

I'm trying to read papers on numerical ode algorithms and I always seem to stumble upon huge amounts of exponentials multiplied by each other. For example in https://arxiv.org/pdf/1208.0689.pdf there ...
2
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1answer
47 views

What is ABA and BAB schemes when talking about numerical integrators

I have read a lot about numerical integrators (ode solvers) lately and tried reading a few papers but I have stumbled upon something that I can't understand and it's something called ABA and BAB. ...
2
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1answer
42 views

Efficient approach for solving matrix plus diagonal matrix system that varies in time

When solving a system of ODEs, as part of a preconditioner, I get the system $(A + D(t))x = b(t)$ where $A$ is a sparse matrix and $D(t)$ is diagonal. I'm currently solving this by taking the LU-...
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1answer
74 views

Actual global error vs theoretical global error: How to combine theory with practice

I have implemented an Adams Bashforth 4 method to solve an Initial Value Problem for an ODE and I am testing it against the test equation: $y'=\lambda y$ with $y(0)=1$ with the exact solution: $y(t)=...
2
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1answer
121 views

Mass Matrix and how to handle it (ODEs) - References

I'm interested in a good reference/paper about how to handle numerically a mass-matrix system as \begin{align} \mathbf{M}(t,y)\dot{y} =F(y,t) \end{align} I know that such a problem can be solved by ...
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0answers
32 views

Discrete maximum principle for discretized ODE

I discretized the following ODE using central finite differences for 1st and 2nd derivatives: $$u''-bu'=f(u), x\in (0,1)\\u(0)=1, u'(1)=0\\ b>0, f:\mathbb{R_{\ge 0}}\to \mathbb{R}_{\ge 0}$$ The ...
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0answers
17 views

Non linear Parametric BVP with inequalities

Consider a non linear ode in dimension $10$: $\dot x = f(t,x,\lambda)$ where $\lambda$ is a vector of $p$ parameters. Consider a boundary value problem of the form : $\dot x(t) = f(t,x(t),\lambda)$ ...
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1answer
47 views

Adam Bashforth 4 method: how to determine starting values and stil keep the the order of accuracy

I am using an Adam Bashforth 4 method to solve an IVP problem so I need other numerical method to estimate the first 3 values. I am very much interested in finding a way to estimate the first 3 values ...
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1answer
47 views

Combining multiple coupled 1st order equations in python

I'm having serious troubles with solving translating 3 coupled differential equations into python. The 3 DE's stem from a 4th order DE used to calculate the bending moment of an underwater pipeline ...
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1answer
60 views

Unexpected solutions solving an ODE using odeint

I am trying to solve a system of 8 coupled differential equations using scipy's odeint. I have already written my code and it runs fine, but the solutions I get are completely different from what I ...
3
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1answer
46 views

Numerical solving Lotka-Volterra ODE in R

Aim: I am trying to numerically solve a Lotka-Volterra ODE in R, using de sde.sim() function in the sde package. I would like to use the ...
1
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0answers
59 views

SDE solver in python: manual determination of integrator step size (dt)

Aim: I am trying to solve a system of SDEs, while using the SDEint package in python 3.x. It is a system of SDEs adapted from and inspired by the Zombie Apocalypse ...
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1answer
87 views

calculating integral for an ODE system

I have an ODE system defining a mathematical model of a biological system, say $$ \frac{da}{dt}=f_1(a,b,\ldots,z,p)\\ \frac{db}{dt}=f_2(a,b,\ldots,z,p)\\ \cdots\\ \frac{dz}{dt}=f_n(a,b,\ldots,z,p) $$ ...
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1answer
62 views

Approximation of ODE solution using Taylor series methods

This is my first post on here, so please excuse mistakes if any. I am trying to plot out the difference between two ODE solvers based on Taylor series: 1st order acccurate: $x(t_0 + h) = x(t_0) + ...
3
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1answer
108 views

Derivation of backward differentiation formulas(BDF)

I have been reading upon numerical techniques that are used to solve stiff ordinary differential equations. From the description given here, I could follow the steps till equation (5). I am finding ...
1
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1answer
37 views

Wrong results for $2$ stage multistep method $y_{n+2} - y_n = h\left[(1/3)f_{n+2} + (4/3)f_{n+1} + (1/3)f_n\right]$

I need to fix a code to utilise the $2$ stage multistep method : $$y_{n+2} - y_n = h\left[(1/3)f_{n+2} + (4/3)f_{n+1} + (1/3)f_n\right]$$ Since this is an implicit method, I used a Newton-Raphson ...
4
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0answers
65 views

Solution of constrained system of ODEs

Can someone point me in a direction to solve this kind of integral constrained system of ODEs. \begin{align} &\int_0^{1/2}\dot{y}^2(t)=p\\ &2\lambda_1\ddot{y}(t)+\pi cos(\pi y(t))=0\\ &y(...
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1answer
63 views

Scipy Two-point Boundary value Problem

Nonlinear ODE Statement I would like to use scipy to solve the following: u'' + (u')^2 = sin(x) u(0)=0, u(1)=1 where u = u(x). Approach I am looking at the ...
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1answer
81 views

Shooting method implementation

I have a trouble of defining a Jacobian matrix for my problem. Basically, I have 4 differential equations to be solved. $$ \begin{aligned} \dot x_1(t)&=x_2(t)\\ \dot x_2(t)&=p_2(t)−\sqrt 2 ...
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1answer
116 views

ODE: should Euler implicit be more accurate than Euler explicit for a given computational step?

I am aware than Euler explicit is conditionally stable, and Euler implicit is unconditionally stable. And I am aware that it is probably pointless to use Euler implicit with a small computational step ...
5
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1answer
106 views

Numerical solution of two coupled nonlinear eigenvalue problems

I would like to numerically solve the following system of coupled nonlinear differential equations: $$ -\frac{\hbar^2}{2m_a} \frac{\partial^2}{\partial x^2}\psi_a + V_{ext}\psi_a + \left( g_a |...
2
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1answer
192 views

Step-size selection for an Trapezoidal Method ODE solver (ode23t)

I was reading the documentation of the MatLab ODE solver ode23t, and I've seen that the trapezoidal rule is used. Moreover, the error is estimated by ...
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0answers
87 views

How to solve an implicit ODE with forward Euler?

Consider the implicit ODE $$ M(y)\dot{y} = F(t,y) $$ If $M$ is non-singular for all $y$ How to use the forward-Euler method to numerically solve for $y$ without inverting $M(y)$? I only came out ...
2
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3answers
138 views

How well do explicit Runge-Kutta “tableau” methods compare to the state of the art ODE solvers and when do they fail?

How well do explicit Runge-Kutta "tableau" methods compare to the state of the art ODE solvers and when do they fail? I've been reading Butcher's ODE book and he does a good job at introducing ...
2
votes
1answer
91 views

Euler Method Instability. Why?

I am currently enjoying writing computational codes as a hobby. Right now I've worked out an Euler method and results are pretty good with up to $x=1$. Over $x=1$, instability starts to kick in. May I ...
3
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1answer
93 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
35 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 ...
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0answers
110 views

Integrating direct dynamics form more than 1 second does not give back the correct result

I am trying to simulate a robot manipulator dynamics in SciLab. Basically, I generated a step function that has constant acceleration for half of the time and then the same acceleration but negative ...
2
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0answers
119 views

How to solve Hamiltonian problems in Julia?

Unfortunately, I have not fountd any comprehensive example for Julia's 1.0 DiffEq.jl Hamiltonian problems. I am trying this: ...
9
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0answers
109 views

Are there shortcuts for numerically approximating systems of ordinary differential equations when autonomous?

Existing algorithms for solving ODEs handle functions $\frac{dy}{dt} = f(y, t)$, where $y \in \mathbb R^n$. But in many physical systems, the differential equation is autonomous, so $\frac{dy}{dt} = f(...
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2answers
110 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 ...
1
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1answer
126 views

Can a second-order ODE be “inconsistent” with its boundary conditions?

I am trying to solve a set of coupled, nonlinear ODEs. The only dependent variable is a 1-dimensional spatial coordinate, let's call it $x$. For now, I've managed to approximate away some of the ...
2
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0answers
80 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} &...
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1answer
207 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{...
1
vote
1answer
63 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 ...
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2answers
126 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 ...
3
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2answers
86 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 ...
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0answers
245 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
107 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: $$ \frac{1}{x^2}\frac{d}{dx}\left(Gam \frac{dL}{dx}\right)+L\left(\frac{a^2x^2}{Gam}-m^2\right)=0 $$ ...
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1answer
69 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
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1answer
98 views

Prescribing variables as an excitation in Runge-Kutta method

I am using Runge-Kutta to solve a $3 \times 3$ 2nd order linear ODE $$M x'' + C x' + K x = 0$$ and initial conditions are all zeros. Then I prescribe the 2nd variable to follow a given path. As for ...
3
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2answers
140 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
77 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 ...
1
vote
1answer
109 views

Scipy odeint Unexpected Results

I am attempting to numerically integrate the equation $$\frac{\mathrm{dP} }{\mathrm{d} r}=-\left ( P+\rho\left ( r \right ) \right )\frac{m\left ( r \right )+4\pi r^{3}P}{r\left [ r-2m\left ( r \...
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0answers
59 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 ...
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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 ...
4
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0answers
98 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 ...
2
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1answer
83 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 ...