Questions tagged [nonlinear-equations]

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

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64 views

Incorporating radiation boundary condition at the edge in finite difference

I am trying to solve the 2-d heat equation on a rectangle using finite difference method. I am confused as to how to incorporate non linear radiation boundary condition at the edge. $-k\frac{\partial ...
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51 views

Oscillations when solving parabolic heat equation with FTCS

I'm wondering if someone could help me out, or point me in a direction of how I can understand the following oscillations that occur when I solve the Porous Medium Equation $$u_t = u_{xx}^{m+1}$$ ...
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500 views

Speeding up the solution of a large set of nonlinear algebraic equations in `sympy`

I have a quite large algebraic equation system to solve, the system is so large, I can't post the example here, so I am posting it to pastebin. The sympy.solve is ...
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1k 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 ...
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62 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 ...
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37 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 ...
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119 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}{...
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52 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 ...
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442 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 ...
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60 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 ...
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39 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 ...
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139 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 ...
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56 views

Object-oriented non-linear solving in python

I'd like to build a system in Python, consisting of (broadly speaking) objects which are internally described with (not necessarily just linear) equations, that I can connect with each other - similar ...
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85 views

Finite difference scheme for unconfined aquifer equation

For an unconfined aquifer we have this PDE for the water table position( of course after somehow making the original Boussinesq equation linearized ): $$ \frac{\partial^2(h^2)}{\partial x^2} + \frac{\...
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66 views

How to get a theoretical background in nonlinear, coupled FEM systems

I'm currently developing simulations for coupled, nonlinear, multi-region systems. Basically, I use the Finite Element Method (FEM) to model each physical quantity in each region. The obtained ...
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121 views

Nonlinear 2D thermoconductivity equation(numerical solution) [closed]

I have to write a solver for 2D equation: $$\partial_t u = u^2(\partial_x ^2 u + \partial_y ^2 u)$$ I try to use explicit method: $$\partial_t u = \frac{u_{i,j}^{k+1} - u_{i,j}^k}{\tau}$$ and $$\...
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202 views

Breather solutions of Sine-Gordon Using Finite Differences

I'm attempting to simulate a standing breather of the form $$ u(x,t)=4\tan^{-1}\left(\sqrt{3}\cos\left(\frac{t}{2}\right)sech\left(\frac{\sqrt{3}x}{2}\right)\right)$$ for the Sine-Gordon equation $$u_{...
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92 views

Which numerical scheme should be used?

Trying to find a way to solve exponentially nonlinear elliptic equation with complex source term i manage to know about such schemes like Godunov, Lax-Friedrich, MUSCL. I still seraching for ...
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57 views

elliptic equation with exponential coefficient

I'm trying to solve the following equation $$\dfrac{\partial}{\partial x}\left(e^{au}\dfrac{\partial u}{\partial x}\right) = 0$$ Of course, this equation can be solved analytically. I am trying to ...
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79 views

Three steps of pde numerical solution and nonlinear equation

I'm very new here. I'm trying to solve nonlinear elliptic equation $$ (n(u)u')' = f(u) $$ and face with crucial misunderstanding. As I suppose, the procedure of solving some nonlinear equation ...
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308 views

Simulating a Hill-Type muscle model

I'm trying to do biologically based locomotion animation. I have a Hill-Type Muscle Model (see reference 1) that I've implemented in code. I can find the initial lengths of the tendons, then do a step ...
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178 views

Numerically solving a system of stiff nonlinear PDEs

I am attempting to numerically solve the following: \begin{align} \frac {\partial y_1}{\partial t} &= i(y_2y_3 - y_2^*y_3^*) - y_1 \tag{1}\\ \frac {\partial y_2}{\partial t} &= y_1^*y_3 - y_2 ...
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70 views

Gauss Seidel moving mesh AMR hamilton Jacobi

I was trying to implement a moving mesh algorithm using Gauss Seidel, so: I have a pde like this (Hamilton - Jacobi eq) $\begin{cases}\phi_t+H(\phi_x)=0 & [-1,1]\times [0,T]\\\phi(x,0)=\phi_0 &...
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242 views

Numerical solution of non-linear advection equation other than inviscid burgers

I am solving a non-linear advection equation of the form $u_t + f(u)_x = 0$ where $f(u)$ is a complicated function of $u$. I am solving this equation using a first order fully implicit scheme (...
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343 views

Solving system of constrained linear and non-linear equations in MATLAB

Solving system of constrained linear and non-linear equations in MATLAB I'm solving a FEM problem in MATLAB with use of the direct stiffness method. The problem is now formulated as a system of nn ...
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113 views

Solving a large system of nonlinear equations, where timeseries are the unknown

I am trying to solve a problem, which I find quite hard, like, headache-hard. I have to solve the following set of $M$ nonlinear equations: $$F(X)=\begin{bmatrix}f_1 (X)\\f_2 (X)\\...\\f_M (X)\\ \end{...
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100 views

Application of vector extrapolation methods to convergence to a steady state solution

I'm working on a fluid solver using dual-time stepping and everything works really well, except the convergence in pseudo-time is slow. I'd like to accelerate the convergence. I know multigrid methods ...
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92 views

Time dependent self-consistent equations

I am facing the following problem. I need to solve numerically a set of coupled equations $$i\frac{d}{dt}f_{n}^{(i)}(t) = \left[U\cdot n(n-1) + \mu\cdot n\right]f_{n}^{(i)}(t) - \sqrt{n+1}\Phi_i^{*}\...
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219 views

Minimal surface finite differences problem - Matlab assemble

I face to the following problem: $$(1+u_x^2)u_{yy} - 2u_xu_yu_{xy} + (1+u_y^2)u_{xx}=0.$$ Problem needs to be discretized and assembled. Does anybody know how to proceed in Matlab?
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87 views

Monotonic convergence of Newton's method for boundary value problems [closed]

I’m interested in solving nonlinear elliptic boundary value problems of the type $$ -a\Delta u + f(u) = 0, $$ $$ u|_\Gamma = u_0 $$ by Newton’s method when its convergence is global and monotonic. ...
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535 views

General algorithm to solve systems of symbolic equations

I want to simplify (solve) a system of linear + nonlinear symbolic equations as much as possible. the equations are of random orders, without differentiation. is there a general & well-known ...
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0answers
328 views

How to solve a nonlinear diffusion equation?

Consider a thin film with a perpendicular applied magnetic field $H_a$ in $z$-axis. The nonlocal relation between $H_a$, the self-field $H_\text{self}$ (generated by the eddy current $J$) and the ...
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1answer
164 views

Numerically solving a non-linear PDE

I have this non-linear partial differential equation. $$ \frac{\partial C}{\partial t}=\left(\frac{\partial C}{\partial x}\right)^2+C\frac{\partial^2 C}{\partial x^2} $$ I want to use the finite ...
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2answers
571 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 ...
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1answer
38 views

Why might the time taken to compute the solution of an ODE system over some interval increase non-linearly with increasing size of interval?

Currently, my problem requires me to solve a system a large system of non-linear ODEs (up to ~5000). So far, I have been using scipy.integrate.odeint as my ...
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1answer
61 views

Solve convection-diffusion equation with a non-linear source term

I would like to solve this equation (which is adapted in my case, a plug flow reactor when a reaction occurs): $ \frac{df}{dt} = D \frac{d²f}{dz²} - u \frac{df}{dz} + r(z,t) $ with $r(z,t)= - k f^{n}...
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1answer
36 views

Coupled pdes of the first order

May question is about possible approaches to solve the following system $$ \begin{array}{rcl} \nabla{n}&=&n\,\mathbf{E},\\ \nabla\cdot\mathbf{E}&=&1-n, \end{array} $$ in general with ...
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1answer
110 views

How to use non-dimensional form in open source codes instead of Units

I am using an open source FEM platform, which requires you to convert your equation system to non-dimensional form. So, there are no units specified for the parameters in the problem. If you use ...
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1answer
109 views

NONLINEAR ENERGY MINIMIZATION EXAMPLE

I am learning about FEM methods and nonlinear optimization. I would like to try my nonlinear trust region solver on some simple nonlinear problem. What would be good example to implement for ...
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1answer
728 views

Solving coupled differential equations and Algebraic equation in MATLAB

I want to solve a system of 7 coupled differential equations and 1 algebraic equation in MATLAB with the method of lines. I could do it for each independent equation with some assumptions, but I can'...
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1answer
70 views

Finding roots without knowing much about the function

Consider solving numerically for roots: $( x_0, y_0): f(x_0, y_0) = 0, g(x_0, y_0) = 0$ where you only know that f, g continuously differentiable but the theoretical differentiation is not a ...
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1answer
537 views

Solver for large non-linear system of equations

I am curently using R package nleqslv for solving a non-linear system of equations with 300 variables. I need to scale this to the system with ~50k variables and naturally this does not scale very ...
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1answer
481 views

MATLAB Newton non-linear equation

I have the following non-linear equation: where $w0=0.25,w0=0.5,w0=0.75$. I have to prove that if $k$ is a root, then also $−k$ is a root and that there exists only one $k∈(0,1)$ root, but my MATLAB ...
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2answers
284 views

Systems of nonlinear equations

Consider the nonlinear system of equations $$ (1) \quad qk^2a_1^2E^2+wna_0a_1AE+pnka_0^2a_1E+rn^2a_0^2A^2-rn^2a_0^3A^2+qk^2a_0a_1ABE-qk^2na_0^2E^2=0, $$ $$ (2) \quad nwa_1^2AE+nwa_0a_1AB+2pnka_0a_1^...
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1answer
9k views

How do I extrapolate data from a NON-LINEAR (logarithmic) standard curve in Excel?

I have made a standard curve. The X-axis is logarithmic. The y-axis is linear. I have added a logarithmic trendline (y = -1.546ln(x) + 39.254; R² = 0.9906). How can I re-arrange the equation to ...
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1answer
93 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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1answer
66 views

FDM on nonlinear PDEs

I'm working with a 2D Navier Stokes PDE in the unstabilized version - the equation is a linear equation of the type $\frac{∂u}{∂t} = F(u,t)$. In order to perform time discretization with FDM (finite ...
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1answer
21 views

Python: Getting second output variable from minimizing a computationally intensive function on first outputs

I have a function in python that is quite computationally expensive to evaluate, of the form: ...
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1answer
116 views

Numerically solving nonlinear parabolic stochastic PDEs

For a project I'm doing, I have to numerically solve a nonlinear parabolic stochastic partial differential equation, of the form $$ u_t = u_{xx} + f(u)(u_x)^2 + a(u) + b(u)W(t, x), $$ where primes ...