Questions tagged [nonlinear-equations]

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

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219 views

Numerical methods for coupled stiff PDEs

I'm dealing with a set of nonlinear coupled PDEs that have the form: \begin{align} \frac {\partial y_1}{\partial t} &= y_2y_3 - y_1 \tag{1}\\ \frac {\partial y_2}{\partial t} &= y_1y_3 - y_2 \...
3
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1answer
191 views

Relaxation Parameters for Steady Navier-Stokes

I am working on a project involving steady solutions for the Navier-Stokes Equations. In the past I've only worked with the unsteady Navier-Stokes, so some of this is new to me. In particular, at ...
3
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1answer
2k views

Solving a Nonlinear BVP using Finite Difference Method

I am trying to write a code to solve a nonlinear BVP using the Finite Difference Method. The BVP is: $(T^2)\frac{\partial^2 T}{\partial x^2} + T \left( \frac{\partial T}{\partial x}\right)^2 + Q = 0$ ...
3
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1answer
220 views

Time Integration of a nonlinear reaction-diffusion system

I want to solve the following system of nonlinear reaction-diffusion equations (Schnakenberg Turing) using FEM methods (such as deal.ii): $$ \partial_{t} u = \Delta u + \gamma\left(a-u+u²v\right)$$ $$...
3
votes
1answer
226 views

Switch branch in bifurcation

I have a system of nonlinear equations $F(x,a) = 0$ and I know that at a specific point $a_c$ a bifurcation occurs, thus the Jacobian becomes singular. How can I switch branches and start following a ...
3
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1answer
89 views

How to pick a basis for the result of a non-linear function given a basis for its argument

I am trying to represent the result of a non-linear function in a small basis, given another small basis that does a good job a representing the argument of the function. More specifically, there ...
3
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0answers
122 views

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 ...
3
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0answers
80 views

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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92 views

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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0answers
68 views

Conjugate Gradient for nonlinear equation system

Is it possible to apply adaptions of the conjugate gradient algorithm i.e. Fletcher-Reeves, Polak-Ribere or others to systems of nonlinear equations? How should the equation system be adjusted so one ...
3
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1answer
57 views

Nonlinear least squares resolution matrix

For a linear least squares problem, it is possible to define a resolution matrix, relating the estimated model parameters to the true model parameters. If we are solving a regularized problem, $$ \...
3
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0answers
70 views

Computation of Troullier-Martins pseudowavefunctions

The computation of Troullier-Martins pseudowavefunctions has been described in [1]. The pseudowavefunction $R^{\textrm{PP}}_l$ is defined by $$ R^{\textrm{PP}}_l(r) = \left\{ \begin{array}{ll} R^{\...
3
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0answers
81 views

Large-scale optimization of nonlinear equations

I'm looking to find a computationally efficient solution to a large system of nonlinear equations. I'm trying to maximize the following function: $$ f(\vec{x}) = \sum_i^N C_i (x_i-A_i)x_i^{\epsilon_{...
3
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0answers
33 views

Detecting blocks in non-linear system of equations

When solving systems of non-linear equations using Newton's method, it is often observed that the system has an independent sub-system, e.g. : $$ f(x,y) = 0 $$ $$ g(x,y) = 0 $$ $$ h(x,y,z) = 0 $$ If ...
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0answers
86 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
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0answers
302 views

How to avoid the Broyden's jacobian approximation becoming poorer with the number of iterations?

I have to solve many times a nonlinear system of the form $$f(x) = b^{(n)}$$ inside a loop. The function $f$ is expensive to compute and I do not have its jacobian, so I have tried the good Broyden's ...
3
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0answers
191 views

Solving a nonlinear poisson equation via variational minimization

I am kind of new in finite elements and I am solving simple "Poisson nonlinear" problem. $- \nabla ((1 + u^2) \nabla u) = f$ $u = 0 \ \text{on} \ \Omega $ I am using Newton solver, where I have ...
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102 views

Applications of algorithm for solving systems of equations with uncertainty

We have been developing algorithms for detecting "robust" zeros of multidimensional functions $f: X\to\Bbb R^n$ where $X$ is an $m$-dimensional domain in $\Bbb R^m$. More precisely, for a given $f$, ...
3
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0answers
188 views

Integration of nonlinear PIDE via spectral methods

At the mean-field level, the dynamics of a polariton condensate can be described by a type of nonlinear Schrodinger equation (Gross-Pitaevskii-type), for a classical (complex-number) wavefunction $\...
2
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2answers
245 views

How can i solve this first order system of differential equations

Edit: I am trying to solve $$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\nu\frac{\partial^{2}u}{\partial x^{2}}\\ u(0,t)=0\nonumber \\ u(1,t)=0\nonumber \\ u(x,0)=\sin(\pi x)\\ 0<...
2
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2answers
157 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}} = \...
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2answers
415 views

2d Euler manufactured solutions

Where can I find manufactured solutions for the 2d Euler equations, with the complete analytical terms, including the Jacobian of the source term ?
2
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1answer
424 views

GMRES vs Newton-GMRES for Solving nonlinear PDE's

Often when numerically solving nonlinear PDE's using method of lines approach with an implicit integrator a system of nonlinear equations have to be solved. To be more specific, let's say we have ...
2
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3answers
123 views

Solving ODE with multiple equilibriums

Consider an ODE of the form: $$ u'(t)=-\frac{1}{\varepsilon}u(u-\frac{1}{2})(u-1) $$ with the initial value $$ u(0)=u_0. $$ Here $\varepsilon>0$ is a constant. It is easy to verify that $u\equiv0$ ...
2
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1answer
148 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)\...
2
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2answers
90 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 ...
2
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1answer
1k views

Numerical Solution to Rayleigh Plesset Equation in Python

I have been trying to numerically solve the Rayleigh-Plesset equation for a sonoluminescence bubble in Python. You can read about this phenomenon here: https://iopscience.iop.org/article/10.1088/0143-...
2
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1answer
757 views

Stopping criteria in iterative methods for solving nonlinear equations

Is it a good criterion to stop iterative methods for solving non-linear equations, such as Newton-Raphson and good Broyden's methods, when $|x_k-x_{k-1}|<|x_k|\,reltol + abstol$ OR when $|f_k|<...
2
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1answer
513 views

Pseudo Code for non linear power function fit needed

I am struggling finding pseudo Code for a non-linear fit of the following function: $y = a\, x^b$ Package NLS in R does perform well, but utilizing external software is not practicable in my program ...
2
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1answer
287 views

How to use Newton-Raphson method to handle nonlinear terms in coupled system of PDEs?

I'm trying to solve the Nonlinear Schrodinger's Equation (NLSE) in 2D using Finite Elements, but I don't know how to handle the nonlinear term. I suppose I have to apply the Newton-Raphson algortihm ...
2
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1answer
192 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 ...
2
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2answers
249 views

Problems where SPD linear system arises

I know some of the places where SPD linar systems arises such as elliptic PDEs and normal equations. Can I have a more comprehensive list of scientific applications which require solving SPD linear ...
2
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1answer
195 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 ...
2
votes
1answer
3k 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+...
2
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1answer
98 views

High order time splitting methods

There are lots of higher order time splitting method as shown by the list with real and complex coefficients $a_i, b_i, c_i$: $$ [e^{c_s \Delta t \hat C}] e^{b_s \Delta t \hat B} e^{a_s \Delta t \...
2
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1answer
74 views

Solving $\sum_r \frac {\mathrm B_{z_r}(a+m_r+1,b)}{\mathrm B_{z_r}(a+m_r,b)}=K$ for $a$ and $b$

How to numerically find the solution for $a$ and $b$ of this equation $$\sum_r \frac {\mathrm B_{z_r}(a+m_r+1,b)}{\mathrm B_{z_r}(a+m_r,b)}=K$$ where $m_r$ are non-negative integers, $0<K<1$, $...
2
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1answer
6k views

Nonlinear ODE to solve Duffing's equation

I am trying to solve the Duffing's equation in MATLAB. $ m\ddot{y}+c\dot{y}+ky+k_{3}y^{3} = f(t) $ where $ f(t) = A \sin{\omega t}$ To do that I wrote a function to be given to the ode45. ...
2
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1answer
141 views

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(...
2
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2answers
177 views

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 ...
2
votes
1answer
146 views

Solution method of nonlinear heat transfer analysis

The governing equation of transient heat transfer analysis is described as follows: $$C \frac{dT}{dt}+K T = Q$$ When using backward difference scheme for the discretization of the time we get the ...
2
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1answer
183 views

Solving this system of equations numerically

In a personal project of mine, I've derived a couple of equations. $$ \sum_{j=1}^Nu_{ij}^*(\theta_1,...,\theta_N,J) - k_{i} = 0 \qquad \forall 1\leq i \leq N$$ $$\sum_{i<j}^N\big(u_{ij}^*(\...
2
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1answer
275 views

Numerical Methods for Solving a Fully Nonlinear Time-Dependent PDE?

Are there numerical methods of solving the following fully nonlinear time-dependent PDE: $$\nabla^2u\left(\textbf{r}(t), \dot{r}(t), t\right)=f\left(\textbf{r}(t), \dot{r}(t), t\right),$$ for $\textbf{...
2
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1answer
99 views

Methods to solve this equation on finite fields?

Is there any analytical (exact, closed-form solution) or numerical method to solve an equation such as $p(x) = r^x$ where $p(x)$ is a polynomial whose coefficients are drawn from a finite field, ...
2
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1answer
146 views

Residual value goes to NaN while solving a system of nonlinear equations

I am solving a system of coupled nonlinear equations using Newton's method, similar to $$\begin{split} c_A(A, B)\partial_tA&=\nabla\left(k_A(A, B)\nabla A\right) + f_A(A, B, t)\\ c_B(A, B)\...
2
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1answer
130 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
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1answer
147 views

Newton's method stagnates at small error

I have a system of the form $$A(u)f(u)=b$$ where $A$ is basically a matrix originating from the Finite Element Method. I try to solve it using the Newton method: $$R = A(u_{i}) f(u_{i}) - b $$ $$...
2
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2answers
88 views

Finding quick solution to a collection of systems of fairly simple but nonlinear equations

So I have a collection of systems of equations, basically $n$ systems of equations, each composed of $k$ equations: $$\frac{a_1x_{1j}}{a_1x_{1j} + \cdots + a_kx_{kj}} + \log x_{1j} + 1 - B_{1j} = 0$$ ...
2
votes
2answers
110 views

numerical approach for system of non-linear partial-ordinary differential equations

I am interested in the numerical solution of the following system of non-linear partial-differential algebraic equations, where the independent variables are $X$ and $T$, representing non-dimensional ...
2
votes
1answer
1k views

Implementation of Backward-Euler scheme, Newton-Raphson iteration scheme to time dependent nonlinear differential equation

I just knew how to do Newton-Raphson iteration in time-independent 1D nonlinear differential equation. Then I applied to time-dependent 1D nonlinear differential equation, and I got confused. Below ...
2
votes
1answer
316 views

How to use time delays in the solvepde function in MATLAB for a system of PDEs?

The solvepde function was introduced in MATLAB R2016a. I am able to solve my system of PDEs if there are no time delays involved. Does anyone know how to include time delays in the solvepde function?

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