Questions tagged [nonlinear-equations]

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

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Splitting system of equations into linear and nonlinear part and solving separately

I was working on a problem recently (calculating all flows in a network given input and output flows, basically what Hardy-Cross tries to achieve) which can be formulated as a well-determined system ...
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Numerical computation of Lyapunov exponents: how to find convergence or non-convergence efficiently?

I am wondering what are the standards for convergence of Lyapunov exponents (and Kaplan-Yorke dimension)? For example, I have a MATLAB code to calculate Lyapunov exponents for the classic Lorenz ...
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2 votes
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Time Integrators for Water Wave Simulation

I am interested in using a numerical wave tank (NWT) to study the performance of various water wave dampers using MATLAB. I am looking at nonlinear water waves. My current NWT (2d, periodic) is based ...
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Numerically solving a 6th order non-linear differential equation in Matlab

I've posted yesterday a question about solving a non linear equation : it was not clear so I am reformulating my question. I am trying to solve a high-order non linear differential equation presented ...
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Easy way to perform solver over pandas dataframe

I'm moving from Excel to Python and I'm trying to solve these equations: $$\begin{align} X_1&=\bigg[\big(3.47-\log(X_2)\big)^2+\big(\log(c)+1.22)^2\bigg]^{0.5}\\ X_2&=\frac{a}{101.32}\bigg(\...
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Preserving conservation properties across time-integration schemes

I am interested in deriving an accurate, implicit discretization of a nonlinear advection-diffusion equation $$ \partial_t f = \frac{1}{v^2} \partial_v J(f,v) \equiv F(f,v) \tag{$\star$} $$ with flux ...
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Fourier-Fourier-Chebyshev 3D non-linear Poisson solver in cartesian coordinates

I am studying turbulence and looking into stuff like Kelvin–Helmholtz instability in plasma. My "current closure" equation is in 3D that I am solving is on the form of generalized non-linear ...
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Global convergence behavior of several Krylov solvers in scipy.opt

In a context of mechanical simulation, where I solve the stationnary action principle directly (i.e. $\nabla S = 0$ for some scalar fonction $S$), I use the wrapper scipy.optimize.newton_krylov to ...
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Why does the total Lagrangian and updated Lagrangian formulations gives different results?

I'm studying Finite Element Procedures by K.J. Bathe and I have trouble understanding how the the total and updated lagrangian formulations should give same equations. Taking a simple truss element as ...
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Why do we solve non-linearity in hyperbolic PDEs that way?

I am used to solve parabolic or elliptic non-linear PDE and the common methods to tackle non-linearity are Picard's iteration and Newton's method. I am a bit confused by the way things are done with ...
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Fitting data with a Voigt function

I have some data, (xrd data), that I would like peak fit with a pseudo-Voigt function, a combination of a Gaussian and a Lorentzian function. These are the functions $G(x) = I \exp\left( -\frac{4\ln(2)...
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Method to linearize highly nonlinear partial differential equation

I have a set of coupled pdes which I want to solve using finite-difference, of which one is nonlinear. The three linear pdes for quantities $T_f$, $T_s$ and $c$ are convection-diffusion-reaction-like ...
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Best way to solve system of quadratic forms

I have a system of equations that have the following structure. Let $x\in\mathbb{R}^m$ and let $x_k$ be the $k$-th element of $x$. Let $H_k\in\mathbb{R}^{m\times m}$ for $k=1,\ldots, m$. I need to ...
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Is it possible to use a fixed point iteration for solving this nonlinear system?

Consider the following differential equation \begin{align} \frac{\partial f(u)}{\partial x} &= g(x), \ \ x\in [x_{L},x_{R}] \label{Eq2.2} \\ u(x_{L}) &= g_{1} \end{align} where $f(u)$ is a ...
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5 votes
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Analysis of nonlinear finite element methods

I have been doing a lot of reading on the development of finite element methods and their analysis using, e.g., functional analysis. I am clear on the formulation of the weak form of a PDE and ...
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Verification of coupled system of equations for light propagation

I am trying to simulate the propagation of light in material using the non-linear schrödinger equation (NLSE): $$\partial_zE=\frac{i}{2k_0}\nabla^2_\perp E+\frac{ik_0n_2}{n_0}\vert E\vert^2E-0.5\beta^{...
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How to treat nonlinear radiation term in heat equation using Finite-element method?

I am trying to solve the time-dependant non-linear heat equation with radiation. This equation is coupled to the radiative transfer equation but for the purpose of my question, this does not matter ...
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Discretizing Multi-species Ion Exchange Equations by Finite Volume Method

I'm solving a system of multispecies ion exchange equations (diffusion+drift fluxes) in 1-d spherical domain using finite volume method to obtain the ion concentrations at the next time step. After ...
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Strange Picard iteration

I am interested in solving the equation $$ \begin{aligned} \nabla \cdot\left(\nabla \phi-\frac{\nabla \phi}{|\nabla \phi|}\right) &=0 & & \text { in } \Omega \\ \phi &=0 & & \...
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How to discretize a non-linear PDE with boundary conditions and intial value

Consider this non linear PDE: $$u_t + c(u^2)_x =\alpha u_{xx} \text{ , } -1<x<1 , t>0 $$ with $$u(-1,t) = g_L(t) \text{ , } u(1,t) = g_R(t) \text{ and } u(x,0)=f(x) $$ where the 3 functions(...
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How does the error work for the Strang Splitting?

We know in Strang splitting that the splitting error in the steady state solution is proportional to $h^2$. I want ask 2 things: If this error in the steady state solution is the global error? If we ...
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1 answer
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Discretization of a non-linear ODE using FDM isn't grid indepenent

I am trying to solve the ODE : $\frac{d^2T}{dx^2} = \omega_1 T+\omega_2 T^2$ + using different numerical methods. I have tried the following discretizations so far and none of them seem to be grid ...
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Approximation in the derivation of the Arc Length method

I am studying the proof of the Arc Length method in section 2.2 of this thesis. In equation (2.2) the author introduces the supplementary conditions $$ (\Delta {\bf u} + \delta {\bf u})^T \cdot (\...
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Solving coupled PDEs with self-consistency condition

I am figuring out how to attack a problem (the Usadel equations of superconductivity) in which I need to solve a set of nonlinear PDEs for the fields $\{G_i (r)\}$ $$ U(G_i(r), \nabla G_i(r), \Delta(r)...
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Stochastic differential equation system (SDE) : overflow encountered in double-scalars

I'm trying to integrate the following SDE system from Dekker et al. [1] $$\begin{cases} \frac{dx}{dt}=a_1x^3+a_2x+\phi+\zeta_x\\ \frac{dy}{dt}=b_1z+b_2(\kappa(x)-(y^2+z^2))y+\zeta_y\\ \frac{dz}{dt}=...
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Solving Laplace equation with constraint on boundary

I have found the following PDE problem in a paper: Essentially, we have a rectangular domain where there is a unknown interface $z=\xi(x)$ (liquid-air interface) separating the domain into two medium ...
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2 answers
133 views

diffusivity matrix assembly in nonlinear finite element analysis

I want to solve a diffusion analysis using finite elements. According to fick's law, governing equation is $$\frac{\partial h}{\partial t} = D \frac{\partial^{2} h}{\partial x^{2}}$$ . h is relative ...
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Linearising Nonlinear Coupled Partial Differential Equations - Alfvénic Diffusion

I am trying to solve the following coupled partial differential equations with a finite difference scheme: $$\partial_tf+v\partial_zf+\partial_z\frac{1}{W}\partial_zf=0$$ $$\partial_tW+v\partial_zW-\...
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Improve code of logarithmic quantizer

I am implementing a logarithmic quantizer which is defined as follows: $$ q(u) = \begin{cases} u_i , \frac{u_i}{1+\delta} < u < \frac{u_i}{1-\delta} \\ 0 , 0 \leq u \leq \frac{u_o}{1+\delta} \\ -...
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3 votes
1 answer
164 views

Nonlinear root solving libraries which accept a Jacobian in band-storage

I'm in search for a library for solving large systems of non-linear equations, similar to MINPACK, but unlike MINPACK, can accept a Jacobian in band-storage. My Jacobian is sometimes not invertible, ...
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4 votes
1 answer
452 views

Backwards Difference Implicit Method for Nonlinear Parabolic PDE in Python

Original Stack Overflow Question: https://stackoverflow.com/questions/65683788/indexerror-index-31-is-out-of-bounds-for-axis-1-with-size-31?noredirect=1#comment116218335_65683788 PDE: u_t = u_xx + u(...
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4 votes
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Continuous limit and nonlinear functional analysis

I have a kind of general question about approximation schemes in nonlinear functional analysis. Given a nonlinear map $\Phi$ from an open set (in an infinite dimensional Banach space) of functions to ...
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1 answer
816 views

Solve non-linear equation in R

I need to solve the following equation for $x$ in [0, 1]. Assume $0<\alpha<1$ and $0<\lambda$. $$(1 - x)^{\alpha+1} - \lambda (x+1)^{\alpha+1} = -2\lambda (\alpha + 1) x^\alpha$$ Would very ...
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3 votes
1 answer
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How to derive the simplified Newton iteration in the TR-BDF2 ODE integration scheme

The Problem The TR-BDF2 explained in this paper [1], 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} &...
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1 vote
0 answers
279 views

Optimize speed for calculating the approximate inverse of a large matrix

I am searching for a faster method to calculate an approximate inverse of a large matrix (up to 32000x32000) resulting from a discrete non-linear system of partial differential equations. I'm using C++...
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2 votes
2 answers
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Numerical Methods of solving a non-linear ODE?

I want to solve the nonlinear equation $\frac{d^2x}{dt^2} + k\sin x = 0$, numerically. I found that solving this elliptic integral would be cumbersome, so is there a numerical method i could use to ...
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5 votes
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81 views

2nd-order TVD criteria for flux-limiter

Consider a nonlinear hyperbolic conservation equation: $$ \partial_{t}u = -\partial_{x}f(u) $$ The spatial derivative of $f(u)$ may approximated after a spatial discretization by $x_{j}=j\Delta x$ $$ \...
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1 vote
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101 views

Numerically estimate the Jacobian from a scalar time-series

I'm trying to numerically estimate the Jacobian from a time-series. Following the paper, https://journals.aps.org/pra/abstract/10.1103/PhysRevA.43.2787. Consider that I have a scalar time series $x = (...
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1 answer
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Interpreting multivariable root-finding results from Matlab's fsolve algorithm

Edit: So I was able to get the same value of r that's given, when coding up the sum of squares of function values directly in the script file, rather than on the Command Window. So, maybe there's a ...
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2 votes
1 answer
173 views

Solving system of nonlinear vector functions

I am trying to figure out how to implement a solver for a system of nonlinear equations of the form \begin{align*} u_1 &= y_n + h\left(a_{1,1}f(t_n + c_1 h, u_1) + a_{1,2}f(t_n + c_2 h, u_2)\...
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1 vote
1 answer
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How to effectively find a starting point solving a non-linear equation?

I have the following equation (the Kurz-Giovanola-Trivedi model [1]) $$ v^2 \frac{\pi^2 \Gamma}{P^2 D^2} + v \frac{mC_0(1-k)\xi}{D[1-(1-k)Iv(P)]} + G = 0, $$ where $Iv(P)=P \cdot \exp(P) \cdot E(P)$, $...
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Derivative-free ill-conditioned non-linear least squares

I am looking for a package which can solve (non-linear) least squares problems without the use of derivatives (because of an expensive model), but which also deals with ill-conditioning well (such as ...
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1 vote
2 answers
867 views

Jacobians with automatic differentiation

I have an objective function F: Nx1 -> Nx1, where N>30000. There are many sparse matrix/tensor multiplications in this function, so taking an analytic Jacobian by paper and pen is cumbersome. ...
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3 votes
1 answer
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How avoid square shape with Laplacian operator in reaction diffusion calculations?

I have used different variants of the Laplacian operator (div grad) using 4, 8, 12, 20 and 24 of the closest points. I get problems due to the chosen coordinate system and the discretization of the ...
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1 vote
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RK4-method starts oscillating above certain input parameters

I am trying to solve an equation of the following type $$\partial_zE(z)=-c_0J$$ with $$J=c_1\beta E^3(z)$$ using the boost::odeint-framework and a fixed time stepper, with $c_0$, $c_1$ and $\beta$ ...
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1 vote
1 answer
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Issue solving nonlinear equation containing a quotient

I have a coupled set of PDEs that need to be solved as part of a larger problem. I am currently approaching this by computing spatial derivatives with finite differences and using PETSc's nonlinear ...
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2 votes
2 answers
104 views

Methods for solving discrete PDEs using algorithmic differentiation results

I'm looking for a method to solve a 20000 variable, 20000 residual non-linear PDE with a Galerkin method. I have Fortran subroutines for: The residuals: $\vec{r}(\vec{x})$; Their Jacobian multiplied ...
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0 votes
1 answer
139 views

Method of Lines Runge-Kutta nonlinear stability and behavior

I have a system of 4 nonlinear 1st-order PDEs. I want to solve them numerically by method of lines, first discretizing space. This leads to the system of $N\times 4$ coupled ODEs. $$ \mathbf{u}_{i} =...
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3 votes
0 answers
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Is the matrix exponential and the Jordan canonical form actually useful for solving differential equations?

All of my yearlong graduate-level Linear Algebra course notes from my professor—an algebraist/representation theorist—shows his love for the exponential map $e^A$ and the Jordan canonical form—and one ...
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2 votes
1 answer
244 views

Numerical methods that can be written in flux conservative form

I have a system of non-linear PDEs that I expect to have shocks as well as the appearance of Gibbs phenomena (spurious oscillations that form near the shock) for 2nd-order methods or higher. I have ...
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