# Questions tagged [jacobian]

Questions related to the calculation or use of the Jacobian matrix or its determinant. Not to be confused with the Jacobi iterative method for solving systems of linear equations. For those, use [iterative-method] instead.

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### Step size constraint in Euler backward

I am dealing with an assignment in MATLAB. It has to do with 'self-driving' cars which are driving in-front/behind eachother. Assuming M cars on a single-lane road, each car adjusts its speed based on ...
430 views

### Number of function calls and jacobian calls in scipy.root

Just as an exercise, I am numerically solving the following system of equations: $$\begin{equation} \begin{cases} x^2 + y^2 = 32 \\ 3x + 7y = 15 \end{cases} \end{equation}$$ ...
47 views

### PETSc non-linear solvers (SNES): specifying single Eval & Jacobian function

The PETSc documentation example of a non-linear solver call has the user provide separate functions for the Jacobian and function evaluations: ...
125 views

### Spot redundant equations within nonlinear system of equations

Is there a general procedure to detect if in a system of m-nonlinear equations (also non polynomial) of n-unknowns some of the equations are redundant? Can the rank of the Jacobian matrix tell me ...
1 vote
31 views

### How to classify ODE equilibria that are stable but slowly changing in value with time?

I'm numerically solving a system of coupled ODEs where time is the independent variable. At each time, I can solve for the equilibrium values of my state variables where their respective derivatives ...
1 vote
92 views

### Computing Jacobian in WENO scheme for advection in a porous media

I am trying to implement an advection equation for a coupled system of a two-phase flow in a porous media using a WENO scheme . My equation is of the form: \begin{align} \frac{\partial (\phi(x,t) C(...
1 vote
379 views

### Calculating Lyapunov exponent (LE) for pendulum using ellipsoid growth - code yields negative LEs

I was redirected here from physics stack exchange where hopefully my question is more appropriate. Per my advisor, I have read the textbook Chaos, an introduction to dynamical systems by Alligood, ...
338 views

### Solving stiff ODEs: Dealing with Jacobian terms which take too long to compute with finite differences

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 ...
192 views

### Why can bad jacobians sometimes works better for implicit ODE method?

I'm solving a system of stiff ODEs describing atmospheric chemistry and transport. I am using CVODE BDF from Sundials Computing. I have two ways to approximate the jacobian: Allow CVODE to ...
256 views

### How do I calculate the Jacobian of this function?

I have two vectors $r$ and $m$. Both vectors are $N\times1$. A function is calculated as - $F(1:N) = \phi r + (r^3 + rm^2)$ $F(N+1:2N) = \phi m + (m^3+mr^2)$ ...
114 views

### Calculating the Jacobian for a function containing a derivative

I have the equation $F(t) = \phi u + \frac{1}{2}\frac{d^2u}{dt^2} + u^3$ and broadly speaking, my task is to calculate the $\phi$ and $u(t)$ such that $F(t) = 0$. I am testing out a new algorithm to ...
729 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 ...
156 views

### Jacobian matrix cutoff in ODE solver

I am studying an implementation of a 3rd semi-implicit Runge Kutta method (siRK3) from the book by Villadsen & Michelsen (1978), Solution of differential equation models by polynomial ...
254 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 ...
1 vote
83 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. ...
Consider the advection equation $$\frac{\partial u}{\partial t}+c(x)\frac{\partial u}{\partial x}=0.$$ With periodic boundary condtitions in $x$ with period $L$, i.e. $u(x,t)=u(x+L,t)$ and initial ...