Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [nonlinear-equations]

Solution of nonlinear systems of equations. The equations might be algebraic or differential equations.

2
votes
0answers
40 views

Nonlinear least squares and regularization

Consider the nonlinear least-squares minimization of a vector of $n$ residuals $\mathbf{f}$ in $p$ parameters $\mathbf{x}$: $$ \min_{\mathbf{x}} || \mathbf{f}(\mathbf{x}) ||^2 $$ This can be done with ...
0
votes
1answer
38 views

Numerical solution of non-linear first order partial differential equation (HJB)

I am trying to solve a simple optimal control problem using the Hamilton-Jacobi-Bellman equation, numerically in Python. This is proving to be rather difficult as I end up having to solve the ...
0
votes
0answers
30 views

Solve system of polynomial equations with Python

I have 5 at most 4th order polynomials in 5 variables, $$p_i(x_1,x_2,x_3,x_4,x_5) \qquad i = 1, \ldots, 5$$ where all coefficients are either rational or floating point. I'd would like to get the ...
3
votes
1answer
32 views

Improve optimization speed for a set of similar problems: Quadratic programming with a warm start

I am repeatedly solving quadratic program, $x^T Q x$ with time dependent linear constraints $Ax=b_t$. Dimension of $x$ is around 10000 and there are around 50 constraints. I want to solve the ...
1
vote
1answer
75 views

Grid dependence of a numerical model

Statement of the problem Suppose, we consider the following model $$ \begin{array}{l} (1)~\mathbf{u}_t + \mathbf{F(u)}_x = \mathbf{S}(\mathbf{u},\mathbf{w}), \\ (2)~\mathbf{w}_x = \mathbf{P}(\mathbf{...
-1
votes
1answer
22 views

Semi-Definite relaxation of non-linear constraint?

I am implementing an optimization problem using semi-definite approach. One of my constraints is of following form $ trace(A∗X)−(k∗trace(A∗X))+(k∗\sqrt {(trace(B∗X)} )==0$ where k is a constant, A ...
3
votes
1answer
66 views

Nonlinear least squares when some parameters are linear

Consider the least squares problem, $$ \min_{\mathbf{a},\mathbf{b}} || \mathbf{f}(\mathbf{a},\mathbf{b})||^2 $$ where $\mathbf{a},\mathbf{b}$ represent the unknown parameters to be found. In my ...
1
vote
0answers
33 views

Has there been a comparison bewteen SIMPLE/SIMPLER and JFNK for steady CFD?

I'm looking for a comparison between the Jacobian-Free Newton-Krylov (JFNK) method performance compared to the conventional CFD nonlinear solution methodologies like SIMPLE. Does anyone know if such ...
7
votes
1answer
161 views

Imbalance of variables in Mixing Newton's method and Linear solver for a Non-linear system

Problem Solving a non-linear system of equations. The number of variables is the same as the number of equations. When I fix a set of variables (say $\vec{y}$) and keep another set free (say $\vec{...
2
votes
1answer
99 views

Prevent single node spikes in a FEM-simulation (using continuous Galerkin)

I am trying to solve a non-linear time-dependent heat equation $$\partial_tT=\nabla \left(k_T(T)\nabla T\right) + f$$ (similar to question Solving a non-linear heat equation with the galerkin method ...
2
votes
0answers
24 views

Neumann boundary conditions in the Maccormack scheme

I am trying to solve the viscous Burger equation $$ \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = \mu \frac{\partial^2 u}{\partial x^2} $$ with Neumann boundary conditions. I am ...
1
vote
1answer
72 views

Multi-steps method for Navier-stokes equations with strongly nonlinear diffusion

I am trying to solve a particular form of the Euler / Navier-Stokes equations in 1D, with very strong and non-linear diffusion coefficients. My system of equations is \begin{cases} \...
2
votes
0answers
65 views

Which SciPy nonlinear solver when Jacobian is analytically known and sparse?

I have a nonlinear function fun with n inputs and n outputs. I also have a function jac which calculates the Jacobian, which is ...
1
vote
1answer
85 views

A reference request for computational plasticity

My background is in applied mathematics and I'm trying to learn plasticity. I have successfully understood the theory and finite element implementation of: linear elasticity, hyperelasticity (Neo-...
2
votes
0answers
134 views

Convergence of a very large non-linear least squares optimization

(note: I also posted this question on stackoverflow before finding this community here, which seems a better place for it) I'm trying to solve the following problem: I have a lot (~80000) surface ...
7
votes
0answers
102 views

Review of modern homotopy methods and practical techniques

I'm hoping someone can recommend recent literature concerning homotopy methods for solving systems of nonlinear equations. Already by the time of Layne Watson's 1986 paper there were a lot of methods,...
1
vote
0answers
23 views

How to model a non-linear least-squares problem for triangles

I have a non-linear least-squares problem to solve and with my current modeling the solver is either very slow or does not converge to a correct solution. For the problem I need to minimize an energy ...
0
votes
0answers
80 views

Non-linear equation solver algorithms

I currently have code that uses the biconjugate gradient stabilized (bicgstab) method to solve $Ax = b$ for $x$, where I never create the $A$ matrix explicitly, but only have a function $F(x) = Ax$ ...
1
vote
1answer
74 views

Approximation of a non linear problem with python

I need your help to solve a problem I'm working on for school. My goal is to approximate the coefficients of a weight matrix so that they check particular properties. So I have $W$ the weight matrix ...
3
votes
0answers
55 views

PETSc SNES for user defined state

How to use PETSc SNES (scalable nonlinear equation solver), when the solution is not a vector but a user defined state? I am implementing a non-linear mechanics problem (geometrically exact shell 5-...
3
votes
1answer
74 views

Solving a non-linear heat equation with the galerkin method gives negative values

I am trying to solve a non-linear time-dependent heat equation $$\partial_tT=\nabla \left(k_T(T)\nabla T\right) + f$$ using the galerkin method, with neumann boundary conditions. For linearization of ...
2
votes
1answer
100 views

Help debugging finite element solution in nonlinear elasticity

I'm writing some code to solve problems in nonlinear elasticity using finite element methods. I have been following Bathe's book but I am having trouble with some nagging details. My question is ...
1
vote
0answers
70 views

Finite difference methods for coupled 2nd order nonlinear pdes

I have a system of coupled nonlinear PDEs that I cannot figure out how to solve in a smart way using FDM, so I was hoping someone here might have a clue. The equations go as: \begin{align*} \frac{1}{...
1
vote
0answers
38 views

Jacobian Elements for Coupled Drift-Diffusion System using Vertex-Centered Finite Volume

I'm trying to solve the fully coupled drift-diffusion system using Newton's Method. Although I eventually plan to potentially use a Jacobian-Free Newton-Krylov approach, this is still something that I ...
4
votes
2answers
168 views

treating “almost linear” nonlinear least-squares problems

As a model for a nonlinear least-squares problem with a large linear part problem, consider $$ \Delta u = 0 \quad\text{in } \Omega,\\ n\cdot\nabla u = 0 \quad \text{on } \Gamma,\\ (u(x_i) - u(y_i))^2 =...
4
votes
1answer
107 views

Nonlinear least-squares solvers vs. generic minimization

A nonlinear least-squares problem with $F:\mathbb{R}^m\to\mathbb{R}^n$, $$ F(x) \to \min_x \quad (\text{in the least-squares sense}) $$ really means minimizing $$ \frac{1}{2} \|F(x)\|^2 \to \min_x. $$ ...
1
vote
0answers
130 views

Alternatives to Newton-Raphson for nonlinear elasticity via finite element

As far as I have seen, solving problems of nonlinear elasticity using the finite element method proceeds by linearizing, either around the initial configuration (total Lagrangian approach) or around ...
5
votes
1answer
115 views

Can redundant variables be beneficial for root-finding convergence

Suppose I have $n$ generally nonlinear equations for $n$ variables, like e.g. for $n=2$ the system $F(x,y)=0$ $$ \begin{aligned} x^2+2y-4&=0\\ \sqrt{8}x+y^2-5&=0 \end{aligned} $$ By ...
1
vote
1answer
48 views

Integrating a nonlinear ordinary differential equation

I am solving an equation of the form $(*)$ $0 = a(f) (\partial_rf)^2 + b(f) (\partial_rf) + c(f),$ where $f$ is a real function of $r\in \mathbb{R}$, and $a,b,c$ are real functions of $f$. The ...
2
votes
1answer
468 views

Solving a system of quadratic equations in Python

I'd like to solve numerically a system of quadratic equations: $A_{11}x_1+A_{12}x_2+A_{13}x_3+B_{12}x_1x_2+B_{13}x_1x_3=C_1$ $A_{21}x_1+A_{22}x_2+A_{23}x_3+B_{21}x_2x_1+B_{23}x_2x_3=C_2$ $A_{31}x_1+...
0
votes
3answers
73 views

Initial conditions for pendulum Jerk equation

I have a very simple problem, but can't seem to understand what I need to do. In simulating a pendulum from it's jerk equation, I'm having a hard time setting initial conditions to get it to work out....
1
vote
0answers
59 views

Rotational kinematics problem @ $\theta = 0$

I'm simulating a magnetic dipole that is subjected to an evolving magnetic field. In ISO convention ($\theta$ is the polar angle and $\phi$ is the azimuthal angle), my equation of motion that is ...
3
votes
0answers
46 views

Nonlinear Sylvester-Like Equation

Maybe you can point me to some results already developed for this. I have to solve for $X$ the following "Sylvester-like" equation: $$ AX - XB = F(X)$$ where $A\in\mathbb{R}^{a\times n}$, $B\in\...
0
votes
1answer
99 views

Non-linearities in modal analysis of flexible beam

I'm trying to analyse the behaviour of a wind turbine blade rigidly connected to a wall during fatigue testing, being excited in it's first mode in two orthogaonal directions simultaneously (the first ...
0
votes
1answer
249 views

Crank–Nicolson method for nonlinear differential equation

I want to solve the following differential equation from a paper with the boundary condition: The paper used the Crank–Nicolson method for solving it. I think I understand the method after googling ...
6
votes
1answer
153 views

Quantifying the degree of nonlinearity in a heat transfer problem

I’m working with a heat equation of the form. $$\frac{d(\rho(T)c_p(T)T)}{dt}-\nabla\cdot(k(T)\nabla T)=f$$ with temperature dependent density $\rho(T)$, specific heat $c_p(T)$, and thermal ...
3
votes
1answer
186 views

Radiation boundary condition (heat transfer)

I am looking for reference on how to implement nonlinear boundary conditions. Specifically, I am interested in implementing a radiation boundary condition for heat transfer with the FEM: $-k \frac {\...
0
votes
1answer
111 views

PDEPE nonlinear

I would like to use Matlab's pdepe to solve this system: $$ s_t =(sr)_x + s_{ xx } \\ r_t =(\frac{ A }{ B }r^2+s)_x + \frac{ A }{ -K } r_{ xx } $$ where $A$, $B$ ...
0
votes
1answer
63 views

Principle of virtual work - extra term needed for deformation dependent loading?

I'm working on a problem in nonlinear elasticity, for which the external forces (loadings) depend on the displacements. Following Klaus-Jürgen Bathe's book "Finite Element Procedures", the virtual ...
2
votes
0answers
147 views

Pros and cons of optimization vs. variational calculus, re: nonlinear elasticity [closed]

I'm trying to solve a problem of finding the displacements of an elastic material subjected to external forces. Those external forces are themselves a nonlinear function of the material displacements....
0
votes
2answers
101 views

Computable alternative to “almost everywhere”

I am working with finite elements for Maxwell's Equations (i.e. with Nedelec's edge elements) and for computation I'm using the FEniCS-project. While implementing the Augmented Lagragian Method, I ...
1
vote
0answers
30 views

Computing Direct Scattering Transform

I'm working on the Nonlinear Schrodinger equation (NLSE) in 1d: $$i\psi _t (t,x)+ \psi _{xx} + |\psi|^2\psi = 0 \, ,$$ for $t\geq 0$ and $x\in \mathbb{R}$. This equation is integrable, and so ...
4
votes
1answer
151 views

Initialize arc length control in Riks' method

I'm trying to use Riks' method and I'm not sure how to set the initial values for the loading coefficient, nor the tangent vector (i.e. the derivative of the displacements and loading coefficient with ...
1
vote
0answers
50 views

Detecting stability in solutions to coupled nonlinear equations in aerodynamics,

I'm studying a quasi-steady force model (published in a fluid dynamics journal) that consists of coupled, nonlinear ODEs that describe unsteady aerodynamics -- recently, my advisor and I have found ...
3
votes
2answers
124 views

How does Mathematica compute real and complex solutions to single, non-polynomial equations?

This question on StackOverflow has led me to ask myself what would be involved in solving an equation like this one using tools available to the Python programmer. $$ \frac{0.125567841}{d^{2.25}} = \...
0
votes
0answers
84 views

1D-Diffusion + chemical reactions: non-inear PDE-System with variable coefficients

I'm modeling a process which involves heat diffusion in 1D as well as different chemical reactions modifying the temperature. The system of PDEs looks something like this: $ \partial T/\partial t=-A\...
0
votes
2answers
278 views

Newton method for a nonlinear system of time-independent PDEs

I have taken a course on undergrad scientific computing which discussed nonlinear algebraic equations about half-way in, and PDEs at the very end, but never discussed nonlinear PDEs. However in my ...
3
votes
2answers
91 views

Approximating solutions to quadratic recurrence boundary value problem

Cross-posted from Math Stackexchange: https://math.stackexchange.com/questions/2421964/approximating-solutions-to-quadratic-recurrence I have a branching process problem that has been reduced to ...
1
vote
0answers
77 views

shallow water equation maccormack method

I am trying to make a code for 1D shallow water equation (nonlinear without source terms) using the MacCormack method for sinusoidal wave propagation. My issue is that the wave fluctuates and does not ...
3
votes
0answers
80 views

Precision not improving by decreasing step-size in nonlinear Schrödinger

I tried to simulate soliton propagation by solving the nonlinear Schrödinger equation using the split-step Fourier method. The following is an example of the Matlab code copied from a textbook. ...