12
votes
Accepted
Why do we solve non-linearity in hyperbolic PDEs that way?
The good thing about the conservative form is that this comprises multiple models, such as Shallow Water Equations, Euler Equations or traffic models.
An essential feature of hyperbolic equations is ...
- 904
8
votes
Accepted
How to solve the problem without using symbolic computation
You can solve this numerically in Python without symbolic computation.
from __future__ import print_function, division
import numpy as np
from numpy import exp
from scipy.integrate import quad
from ...
8
votes
Limitations with dynamical systems vs. PDEs?
PDEs are a form of dynamical system where there is another continuous variable. Usually this is space, so you're looking at how things over time and space instead of just over time.
Here's an ...
- 12.2k
8
votes
Jacobians with automatic differentiation
Julia has a whole ecosystem for generating sparsity patterns and doing sparse automatic differentiation in a way that mixes with scientific computing and machine learning (or scientific machine ...
- 12.2k
7
votes
Solving a linear equation system with pure Neumann condition
Pure Neumann problem is unique up to a constant. My two favourite solution strategies:
Modifying the equation $-\Delta u = f$ to $-\Delta u + \varepsilon u = f$ for some small $\varepsilon>0$. If ...
- 2,041
7
votes
Accepted
Newton iteration applied to nonlinear PDE
It's a bit easier to see if you write your equation in the a semi-discretised system of the form $u^{\prime}(t) = F(u(t))$ and with the application of the $\theta$-method and approximating $u^{\prime}(...
- 5,369
7
votes
Accepted
Understanding the Eisenstat-Walker method for choosing the tolerance of a linear solver when solving a non-linear PDE
$s_k$ is the "approximate Newton" search direction. So in essence, when they say
Choose $s_k$ such that
$\|F(x_k)+F'(x_k)s_k\| \le \eta_k \|F(x_k)\|$
they are saying: Solve the Newton system $F'(...
- 53.7k
6
votes
Accepted
treating "almost linear" nonlinear least-squares problems
If you change variables to optimize for the residual of the linear part, then the Hessian will be a low-rank update to the identity. Then L-BFGS would work very well. Specifically, your problem takes ...
- 3,063
6
votes
Accepted
GMRES vs Newton-GMRES for Solving nonlinear PDE's
The reason is that GMRES can only be used for solving linear equations, i.e. equations of the form $Ax=b$, where $A$ is some matrix and $x,b$ are vectors. What GMRES does, essentially, is it ...
- 106
6
votes
What are systematic ways of approximating a non-smooth (non-continuously differentiable) system dynamic to be n-smooth?
Two systematic ways of smoothing a function $h$ would be:
1. Join the piecewise smooth parts of your function using Hermite interpolation so that the derivatives are matched to your satisfaction.
2. ...
- 3,994
5
votes
Accepted
2d Euler manufactured solutions
This 2004 paper by Roy et. al in Int. J. Numer. Meth. Fluids 2004; 44:599–620 (DOI: 10.1002/d.660) should contain exactly what you're looking for:
ftp://ftp.demec.ufpr.br/CFD/bibliografia/...
- 2,166
5
votes
Accepted
What is the preferred method for evolving the Nonlinear Schrödinger Equation?
You won't find universal agreement among experts on a single best way to solve this equation. On the other hand, since it is a PDE in one spatial dimension there are many approaches that will give ...
- 16.4k
5
votes
Accepted
Pseudo Code for non linear power function fit needed
Chapter 10 of Numerical Analysis (Richard L. Burden, J. Douglas Faires) gives good readable pseudo code for Newtons method. The start parameters are taken from the solution of the linear problem of ...
- 289
5
votes
Accepted
What would be a good approach to solving this large data non-linear least squares optimisation
I don't know much about tracking implicit surfaces, so I'm just going to start with the optimization problem and go from there.
The optimization problem is, at the core, nonlinear least squares, and ...
- 3,063
5
votes
Accepted
Stopping criteria in iterative methods for solving nonlinear equations
It can happen in principle that the condition on $f$ will never be satisfied, if, for example, $|f'(x_*)|u>\mathit{tolfun}$, where $u$ is the unit roundoff.
When I tried your example with $K=10^8,...
- 11.4k
5
votes
How does Mathematica compute real and complex solutions to single, non-polynomial equations?
As the other Answer already touches on the possibility of a symbolic root-solver being applied to this particular equation (by transforming into a polynomial form, albeit of degree $\ge 5$), I'll make ...
- 3,349
5
votes
Accepted
Nonlinear least-squares solvers vs. generic minimization
If we let
$\phi(x)=\sum_{i=1}^{m} F_{i}(x)^{2}$,
we could compute $\nabla \phi(x)$ by finite difference approximation. However, it is generally smart to make use of the special structure of $\phi(...
- 18.5k
5
votes
Discrete-time Algebraic Riccati Equation (DARE) solver in C++
If you want a ten-line solution that is decently fast and stable, you can implement yourself the structured doubling algorithm: set up the coupled iteration
$A_0 = A, G_0 = G = BR^{-1}B^T, H_0 = Q$
...
- 10.6k
5
votes
Accepted
Numerically solving a non-linear PDE
First off, the PDE can be rewritten instead as
$$\frac{\partial C}{\partial t} = \frac{\partial}{\partial x}C\frac{\partial C}{\partial x}$$
or, by applying the product rule in reverse again, as
$$\...
- 9,813
5
votes
Accepted
Solving system of nonlinear vector functions
If you have a single vector equation $\vec{F}(\vec{x})=0$ then you solve it by representing that state vector $\vec{x}$ as a set of amplitudes $[x_0,x_1,...,x_{n-1}]$ after discretization by your ...
- 2,370
5
votes
Spot redundant equations within nonlinear system of equations
In the example you give, the two equations are not redundant. Each of the two equations describes a set of lines in the 2d plane, and the lines happen to be tangential at a specific point -- which is ...
- 53.7k
4
votes
Accepted
Algorithm for solving system of quadratic equations and linear equations
Quadratically constrained quadratic programming (QCQP) focuses on convex inequalities because those preserve the convexity of the problem. Quadratic equalities do not, so this problem is much harder.
...
- 56
4
votes
Solving a linear equation system with pure Neumann condition
Including even one Dirichelt condition changes the problem you are trying to solve and will not give you the correct solution! You must fulfil the Discrete Compatibility Criteria, see e.g. first and ...
- 127
4
votes
Accepted
Methods of solving non-linear advection-diffusion systems beyond Newton-Raphson?
I'm assuming the limitation in 2D and 3D is storing the Jacobian.
One option is to retain the time derivatives and use an explicit "pseudo" time-stepping to iterate to steady state. Normally the CFL ...
- 181
4
votes
Accepted
Evolving nonlinear Schrodinger equation with higher-order algorithms?
Your added edit to the question points exactly the way this needs to be done: You first evolve your field using a half-step with the $\hat D$ operator, then a full step with the $\hat N$ operator, and ...
- 53.7k
4
votes
Solving ODE with multiple equilibriums
There are a couple of questions implicit in your post:
How does one deal with non-uniqueness of the algebraic equations generated by any implicit numerical method?
Typically you have a very good ...
- 16.4k
4
votes
Newton method for a nonlinear system of time-independent PDEs
Conceptually, forget about the fact that the problem is time dependent for a moment. That's because if you apply a time stepping scheme to your time dependent PDE, you're going to get a (in your case ...
- 53.7k
4
votes
Accepted
C++ template design pattern for groups (algebra)
Chapters 6 and 7 of Stepanov and Rose's book From Mathematics to generic programming deal precisely with what you asked. The first author is the designer of the C++ Standard Template Library.
- 3,994
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