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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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0answers
38 views

### Solving PDEs: What is the best way to deal with non-banded/dense jacobians?

I have a system of PDEs describing atmospheric chemistry and transport. I use finite-differences to make my system of PDEs into a system of ~10,000 ODEs. I then integrate the ODEs forward in time with ...
0answers
39 views

### scipy solve_ivp with adaptive solution

I am struggling to understand how scipy.solve_ivp() handles errors in a system of ODE. I have a complex systems of ODE with very large varying time-scales, for which my state "y" is composed ...
0answers
51 views

### Problems with 'ODEint - Excessive work done' and using pySINDy

I am create syntetic data from a system of differential equations describing the COVID-19 epidemic and then I try to use the method SINDy to recreate the differential equations from the data. Link to ...
2answers
75 views

### numerical solution for differential equation

I have these 3 equations and i want to solve them with numerical methods. so I am using scipy library but I don't know how to solve 3 equations together. R, g, sigma and density are constants. \begin{...
0answers
40 views

### A JAVA solver for ODEs with boundary conditions (BVP)

I need to solve a system of linear first order ODEs with boundary conditions in JAVA. I was wondering if any of you know of a JAVA package with the capability of solving a boundary value problems? (i....
0answers
37 views

### Path constraints for state variables - fmincon, ODE45

My problem lies in constraining state variables (look for 23,24,25). I am currently using ODE45 to solve equations and fmincon to find best control variable.How would you go about solving this? Here ...
2answers
527 views

### Specifying ode solver options to speed up compute time

I'm specifying the 'JPattern', sparsity_pattern in the ode options to speed up the compute time of my actual system. I am sharing a sample code below to show how I ...
1answer
87 views

### Preferred application for shooting method

Every now and then there are questions asked in this site related to the shooting method for boundary value problems (see 1, 2, 3). Nevertheless, in some of the cases that I have seen here, the ...
1answer
94 views

0answers
52 views

### Jacobian matrix cutoff in ODE solver

I am studying an implementation of a semi-implicit Runge Kutta method of 3. order (SIRK3) from the book by Villadsen & Michelsen (1978), Solution of differential equation models by polynomial ...
1answer
40 views

### Question about scipy’s ivp solver

there is one thing I don’t understand. Is the tolerance to compute the step size updated at each timestep or fixed at all timestep. Also, when we look at the documentation and how the tolerance is ...
1answer
69 views

### Geometric integrators besides midpoint/Crank-Nicolson?

I have a first-order ODE $$\dot{x} = a(t) \times x, \quad x(0) \in\mathbb{R}^3.$$ with $\|a(t)\| = 1 \;\forall t$. Consequently, $\|x(t)\|=\|x(0)\|$ for all $t>0$. I would like this to be ...
0answers
36 views

### Using event function to shift between 2 sets of ODE

I have a bunch of ODEs I am trying to solve using ode45 in MATLAB. I have hidden the details of the equations to keep it simple (so as to build a general algorithm)! \[\frac{dR}{dt}= F_{1}(R,N) \quad\...
1answer
83 views

### Numerical integrator for $a'(t)=e^{-a(t)}f(t)$

Suppose I know a function $f(t)$ and all its derivatives in $t$ in closed form. Given $a(0)$ and some $t_0>0$, I'm looking for an explicit integrator that can estimate $a(t_0)$, where $a(\cdot)$ ...
1answer
147 views

### N-body problem with differents solvers (RK2, RK4, Euler symplectic, Stormer-Verlet) : planets drift to infinity

I'm trying to write an integrator for the 2 and 3-body problem. I choose to start from a generalisation to N-body problem so I can just pass my bodies to the same integrator in the two cases. I'm ...
1answer
51 views

### finding boundary conditions when transforming a higher order ode to system of first order ode

given the following ODE: $$\frac{d^{4}w}{dx^{4}} + B\frac{d^{2}w}{dx^{2}} = 1$$ with boundary conditions $w(0) =0 , w(1) = 0,w'(0) = 0,w'(1) = 0$ its possible to solve analytically but I am attempting ...
1answer
209 views

### Solving Cahn-Hilliard equation using semi-implicit Fourier spectral methods

So, I have written both a C and python code to solve the 2D Cahn-Hilliard equation: \begin{equation} \frac{\partial c}{\partial t} = \nabla^2\left(c^3 - c - \kappa\nabla^2c\right) \end{equation} ...
1answer
66 views

### ODE45 and a variable that assumes multiple values during the timespan

I have tried in different ways to see what happens to voltage V and gating conductances m, n and h when, at time step x, current I switched from 0 to 0.1, and then at time step x + n it gets back to 0....
1answer
170 views

### Drawing saddle node bifurcation diagram for a non-linear ODE in Python

I'm trying to draw the bifurcation diagram of the following ODE, This ODE leads to a saddle-node bifurcation (see wiki) However what I get is not exactly right. There's a lot of "noise" as ...
2answers
197 views

### Speed up solution of a very large system of ODEs

I need to solve many very large systems of first order ODEs, which describe some chemical reactions. The number of variables (in each system) is on the order of $n \sim 10^5$. I am using ALGLIB vector ...
1answer
147 views

### Perturbation problem using Runge-Kutta 4

I'm trying to evaluate the perturbations magnitude between 2 body orbiting a central one in three dimensions. In order to do this I need to have an estimate of the error, which I did using Richardson ...
0answers
103 views

### Does non-dimensionalizing ODEs affect the stiffness of the equations?

Does non-dimensionalizing ODEs affect the stiffness of the equations? Can it improve the stability of numerical methods like ode45,ode113 in MATLAB? I am trying to solve 2 eqns. which might involve ...
1answer
202 views

### Comparing numerical solutions with very different time grids

I've read an article (Long-term integrations and stability of planetary orbits in our Solar system) in which the authors solved the problem of the absence of an analytical solution for the solar ...
1answer
65 views

### Solving differential equation by specifying jacobian pattern

This is a follow up to my previous question posted here I'm trying to construct the sparsity pattern of the jacobian matrix to speed up the computation of a large system of odes. The following is the ...
1answer
48 views

### solving differential equations with jacobian pattern

I'm trying to compare the simulation time for solving a system of differential equations with and without jacobian pattern for a toy model using ode15s in MATLAB. ...
2answers
97 views

### Numerically solving the equation of motion for inflation in cosmology

I want to solve the equation of inflation involving a scalar field numerically using Python libraries such as odeint or scipy. ...
1answer
74 views

### Maintain unitary time evolution for a nonlinear ODE

I want to solve a nonlinear ODE of matrix $A(t)$ $$\mathrm{i}\dot A = A(t)M(t),\:\mathrm{with}\: M(t)=A^\dagger(t)H(t)A(t)$$ where $H(t)$ and hence $M(t)$ are Hermitian. Therefore, I presume the time ...
2answers
215 views

### Solve a system of coupled differential equations in Python

I have a system of two coupled differential equations, one is a third-order and the second is second-order. I am looking for a way to solve it in Python. I would be extremely grateful for any advice ...
0answers
33 views

### Estimating the Jacobian in Harmonic Balance Method

I am trying to solve a set of ODEs using the Harmonic Balance method. In order to do this, I need to compute the Jacobian of the set of equations. However I am very confused regarding the dimensions ...
0answers
102 views

### “This DAE appears to be of index greater than 1” daeic12 (line76) error code

Hi I am trying to solve a set of pde converted into ODE and DAE using central finite difference method. I have used the MATLAB 'solve' command to determine the coefficients of fictitious nodes for ...
0answers
39 views

### N-body correct scaling

I realized an usual way to scale an N-body problem for an N-body simulation is by choosing units such that gravitational constant $G = 1$, but I'm probably doing it the wrong way. Suppose I simply ...
1answer
101 views

### Numerical solution of ill-conditioned differential equation

I want to solve the following Cauchy problem \begin{equation} y' = y^2 + \frac{t^4 - 6t^3 + 12t^2 - 14t + 9}{(1+t)^2} \end{equation} with initial condition: $y(0) = 2$ for $t \in [0,1.6]$ using a 3 ...
1answer
126 views

### Floating point and global error in Euler Method

Inspired by this answer, I tried to understand when floating point errors come into visibility and to check it also comparing the plot of the numerical solution with Explicit Euler with the analytical ...
1answer
298 views

### ODEintWarning: Excess work done on this call (perhaps wrong Dfun type)

I was messing around with some numerical integration functions. I wrote an arbitrary differential equation to test my understanding, the code is as follows: ...
1answer
368 views

1answer
57 views

### (2) Trying to model a simple second order ODE: Why time-step smaller is not better

This question is related with this other question: Trying to model a simple second order ODE. On this other question, I get some useful comments on why the simulations are so terrible. However, I have ...
2answers
75 views

### Efficient ODE steppers with query of $f$ and $\nabla f$ is efficient

Assume we have an IVP $y'(t) = f(t,y)$, and that $\partial_t f$ and $\nabla f$ are cheap to compute. Assume further that more derivatives are not cheap to compute, or inaccessible for some reason, ...
2answers
282 views

### Trying to model a simple second order ODE

I am studying some computational methods and I am trying to program simples equations to understand how the methods work... Particularly, I am trying to understand how multiorders ODE's work. I've ...
1answer
81 views

### What are the advantages and disadvantages of using norm error control in the MATLAB ODE suit?

In MATLAB's ODE suit, there seem to be two basic methods of controlling the Local Truncation Error (LTE) of the ODE which the user can choose from, namely: The absolute error control (default), ...
4answers
137 views

### Getting torsion and curvature out of ODE solution skeleton

Suppose I have solved an ODE $v'(t) = f(t,x)$ via some adaptive stepper, such as RK4 or Dormand-Prince, generating a list of points $\{(t_i, v_i, v_i' = f(t_i, v_i))\}_{i=0}^{n-1}$. I wish to use this ...
0answers
88 views

### How to derive the adjoint sensitivity equations for a least squares objective function gradient

The Problem I would like to determine the gradient of a least squares objective function which depends on a vector of 40 parameters $p$, and the solution of a system of 32 differential equations. In ...
1answer
133 views

### How to select initial time step in adaptive time step ODE solver (TR-BDF2)

The Problem I am currently reconstructing a TR-BDF2 scheme which contains the following two stages: \begin{align} y_{n+\gamma} & = y_n + \gamma \frac{h}{2}\left( f_n + f_{n+\gamma} \right) \...
1answer
62 views

### If I rescale the time in a differential equation, do I need to adjust the parameters?

Imagine I have a differential equation and I have some data and the model is supposed to fit the data. If I now rescale the time in the range 0 to 1, do I need to adjust the parameters of the ...
1answer
137 views

### How to derive the simplified Newton iteration in the TR-BDF2 ODE integration scheme

The Problem The TR-BDF2 explained in this paper , is quite a popular numerical scheme used to integrate $\dot{y} = f(t,y)$, consistent of the following two stages: \begin{align} y_{n+\gamma} &...
2answers
342 views

### How to set up the differential equation system to speed up computation?

I've set up a system of differential equations, obtained after discretizing pde, in the following way ...