Questions tagged [numerical-analysis-of-pde]
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107 questions
16
votes
4
answers
3k
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Is there a connection between representation theory and PDEs?
As a PhD student, if I want to do something algebraic / linear-algebraic such as representation theory as well as do PDEs, in both the theoretical and numerical aspects of PDEs, would this combination ...
- 179
11
votes
3
answers
678
views
Which matrices can be realized as the Dirichlet-to-Neumann map for a given domain?
Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $n$ electrodes (Dirichlet BC $u=\text{...
- 532
8
votes
1
answer
3k
views
Review paper/book on Finite Difference Methods for PDEs
I am looking for a good, relatively modern, review paper/book on Finite Difference Methods for PDEs with a theoretical emphasis in mind. By theoretical emphasis I mean that I care about theorems (i.e. ...
- 391
7
votes
2
answers
928
views
What are dissipative PDEs?
I often come across the term dissipative (partial) differential equation in mathematical articles, especially in the context of hypocoercivity and entropy methods. I now have an intuitive idea of ...
- 185
6
votes
1
answer
415
views
Difference stencils approximating Laplacian
Let $\Delta$ be the Laplace operator on the interval $[0,1]\subset \mathbb{R}$.
Divide $[0,1]$ into small intervals of size $h$ to get an equidistant grid. One can approminate $-\Delta$ on this grid ...
- 161
6
votes
1
answer
762
views
What is the big-O complexity of solving the sparse Laplace equation in the plane?
In MATLAB, you can get a 2d Laplacian via A = delsq(numgrid('S',N)); yielding a matrix $A$ that is $n \times n$ with $n = O(N^2)$, for a square domain discretized ...
- 981
6
votes
0
answers
392
views
Steklov eigenvalue problem for a planar region bounded by ellipse
The Steklov problem for a compact planar region $\Omega$ is
\begin{cases} \Delta u =0 &\text{in $\Omega$}, \\ \frac{\partial u}{\partial n} = \sigma u &\text{on $\partial \Omega$},
\end{...
- 141
5
votes
1
answer
114
views
Is the minmod limiter energy stable?
It is well-known, that upwind scheme and Lax-Wendroff scheme are energy stable for the linear advection equation $u_t +a u_x = 0$ with periodic boundary conditions, if the CFL condition is satisfied, ...
- 161
4
votes
2
answers
566
views
How to compute $\sin(\frac{d}{dx})f(x)$?
Assuming $f(x)=e^{-x^2}$ for $x$ in $[-10,10]$, I have tried the following:
Fourier transform $\mathcal{F}$: $\frac{d}{dx}$ can be diagonalized as $\mathcal{F}^{-1} i\omega \mathcal{F}$. Therefore, $\...
- 350
4
votes
2
answers
1k
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Representing a nonlinear elliptic PDE as an energy minimization problem
I need to solve a PDE in 2D representing a (time-independent) nonlinear diffusion process. The unknown function is $\phi(x,y)$ and its gradients create fluxes $\vec J$ through a nonlinear relation:
$$\...
- 265
4
votes
0
answers
198
views
Pricing zero coupon bonds through PDE
I'm currently studying Paul Wilmott on quantitative finance and saw an interesting idea for an interest rate model that went unexplored in the book.
The idea is to model the market price of risk as a ...
- 51
4
votes
1
answer
172
views
Numerics for continuity equation with Sobolev vector field
Has any work been done about numerical methods for the continuity equation
$$
\partial_t \rho(x,t) + \operatorname{div} (b(x,t) \rho(x,t)) = 0, \qquad t \in [0,T], \quad x \in \mathbb R^N,
$$
where $...
- 839
4
votes
0
answers
706
views
What does the Von Neumann's stability analysis tell us about non-linear finite difference equations?
I've asked this question on computation science stackexchange, but it did not receive any answers so I have decided to ask it here as well.
I am reading a paper [1] where they solve the following non-...
- 215
3
votes
1
answer
245
views
How to generate a random function with conditions?
The background is as follows:
I consider the following differential equation
$$\phi_{xx}+u\phi=\lambda \phi,\ \ \lambda=-k^2$$
where $u=u(x),\ \phi=\phi(x,\lambda)$, $\lambda$ is the spectral ...
- 33
3
votes
1
answer
292
views
Conditions on the velocity ensuring that a flow moves points along the boundary of a manifold
Let
$\tau>0$;
$d\in\mathbb N$;
$v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ be Lipschitz continuous in the second argument uniformly with respect to the first with $v(\;\cdot\;,x)\in C^0([0,\tau],\...
- 167
3
votes
2
answers
428
views
Inconsistency in determinability of the solution of a linear first order PDE
Consider the following differential equation:
$$\frac{\partial u(x,t)}{\partial t} = - \frac{\partial u(x,t)}{\partial x} + u(x,t) \label{1}\tag{1}$$
with $u(x,0)=f(x)$. The solution of \eqref{1}, ...
- 350
3
votes
1
answer
747
views
Navier-Stokes equations and machine learning
I am looking for a reference explaining how to solve the Navier-Stokes equations numerically using machine learning algorithms .
Thank you in advance for your help .
- 143
3
votes
1
answer
227
views
Solving a differential system
Let $\mu$ be a probability measure on $\mathbb R$ with Lebesgue density, i.e. $\mu(dx)=\mu(x)dx$. We aime to find increasing and decreasing functions $\phi_{+}: \mathbb R_+\to \mathbb R_{+}$ and $\...
- 1,835
3
votes
1
answer
327
views
Numerical iterative methods for Poisson equation
Given a domain $\Omega \subset \Bbb R^n$ and $\Delta\varphi=f$ where $\varphi:\Bbb R^n \to \Bbb R$ is unknown and $f:\Omega\to \Bbb R$ is a blackbox function (for each $\bf x$ it provides $f({\bf x})$,...
- 135
3
votes
1
answer
413
views
Weird claims and conclusions in "Introduction to Shape Optimization"
I'm trying to understand the notions of Euler and Hadamard derivatives of shape functionals. All the lecture notes and papers on this topic that I've found seem to build up on the books Shapes and ...
- 167
3
votes
2
answers
364
views
Questions for the non-linear PDE $2u_t=\log(-u_{xx})$
Consider the PDE as follows :
$$2u_t=\log(-u_{xx}), \quad \forall (t,x)\in [0,1)\times (-1,1)$$
with the terminal and boundary conditions
$$u(1,x)=0,\quad \forall -1<x<1 \quad\quad \mbox{and} \...
- 1,334
3
votes
1
answer
431
views
Prove of the shape-derivative identity relating the shape and material derivative of a shape-dependent function
I've started reading about shape optimization. Most of the concepts I've encountered so far (such as the shape derivatives of domain and boundary integrals and the corresponding) seem to be complex, ...
- 167
3
votes
1
answer
206
views
More important or relevant progress in discretizing hard problems in physics in last decade
This is a reference request, and soft question as companion.
I'm curious to ask, from an informative point of view, what about the more important progress in the goal to discretize hard problems in ...
3
votes
0
answers
99
views
Rate of convergence of mollified distributions in Besov spaces with negative regularity
Given a standard mollifier $\rho_\delta$ and a distribution $ u \in B^\alpha_{ p, p}$ with $\alpha<0$, $p \in [1, \infty]$ and $B^\alpha_{p,p}$ is a not-homogeneous Besov space, I'm trying to prove ...
- 293
3
votes
0
answers
74
views
Confusion with implementation of PDE constraint Bayesiain inverse problem
Consider a PDE,
$$\partial_t u -a \nabla u - ru (1-u) = 0$$
at a given snapshot in time. The inverse problem is to find the diffusion coefficient $a \in L^{\infty}$ from a noisy measurement $$Y = \Phi(...
- 81
3
votes
0
answers
61
views
How I can distibute values over the computational cells?
I am an engineering student and I try to solve the fluid equations over a given set of computational cells. I have a mathematical question about a field I am currently studying, precisely the ...
3
votes
0
answers
61
views
Tuning parameters of PDEs given a set of data
I am interested in doing statistical inference in the context of PDEs. Loosely speaking, the kind of problem I have in mine is the following.
Problem setting
Let $(t_i, x_i, y_i) \in \mathbb{R} \...
- 823
3
votes
0
answers
89
views
What is the purpose of converting a level-set function into a signed distance function?
In the paper Electrical impedance tomography using level set representation and total variational regularization, the authors tried to implement an iterative algorithm to find the interface of two ...
- 161
3
votes
0
answers
121
views
Smoothly connecting PDEs with finite differences
A PDE with non-smooth inhomogeneity
Let $\mathcal{L}$ be a second-order, linear, elliptic differential operator acting on $\mathcal{C}^2([0,2]^2)$.
I'm numerically solving the inhomogeneous PDE
\begin{...
- 255
3
votes
0
answers
86
views
Shooting method still relevant?
I'm preparing to teach a "Numerical Analysis II" course next term, and in previous years this course involved a section on the shooting method for solving one-dimensional boundary value problems. This ...
- 243
3
votes
0
answers
73
views
Spectrum of a symmetric saddle point matrix
Let $C=\left[ {\begin{array}{cc}
A & B^{T} \\
B & O \\
\end{array} } \right]$, where $A\in \mathbb{R}^{n\times n}$ is SPD, $B\in \mathbb{R}^{m\times n}$ and $m\leq n$. The matrix $B$ ...
- 31
3
votes
1
answer
305
views
Methods to compute the Green's function for the 1D wave equation with nonsmooth coefficient?
I am seeking advice on the best available numerical methods to compute the Green's function for a 1D wave equation with rough coefficient.
Suppose that the coefficient $c(x)$ in the 1D wave equation ...
- 115
3
votes
0
answers
244
views
Numerical inversion involved confluent hypergeometric (1F1) (or Kummer function)
Edit: The question is solved !! The code is actually correct. There is not error in the codes. I miss-used it. Thank you for your attention : )
This problem arises when I tried to compute the valua ...
- 323
2
votes
2
answers
328
views
$H^s$ norm of non-integer power of functions
Let $ \Omega = \mathbb{T}^d (1 \leq d \leq 3)$ be the $d$ dimensional torus and $ u \in H^2(\Omega) $ be a complex valued function. For some $ 0 < \alpha < 1 $, let $ g(u) = |u|^\alpha u $.
My ...
- 105
2
votes
1
answer
62
views
Lumped mass matrices and bubble functions for tetrahedral elements
For whatever reason, I stubbornly decided to use tetrahedral elements and find myself needing to use P3 elements with bubble functions ("P3b3d" in FREEFEM-style denomination).
The 2d case is ...
- 981
2
votes
1
answer
139
views
Construct examples satisfying some inequalities [closed]
How do I construct two vectors $a,b\in \mathbb{R}^{n}$, $a=(a_1,a_2,\ldots, a_n)^T$ and $b=(b_1,b_2,\ldots, b_n)^T$ which satisfy in the following conditions
\begin{align}
& a_ib_i\geq 1,a_ib_j&...
- 37
2
votes
1
answer
282
views
Numerical methods for IDE [closed]
I would like to read a popular literature on the topic "Numerical methods for integro-differential equations".
Could you recommend me any articles or book with a brief overview of some methods (maybe ...
2
votes
2
answers
262
views
Iterative method for $p$-Laplacian
Consider the following iterative procedure for solving the $p$-Laplace equation $\nabla \cdot (|\nabla u|^{p-2} \nabla u) = 0$ with fixed Dirichlet boundary data:
$u_0$ is our initial guess, for ...
- 648
2
votes
2
answers
116
views
Solving numerically an equation involving exponentials [closed]
I met an equation of the following form:
$$\sum_{i=1}^nk_ip_i e^{-k_i\lambda}~~=~~b,$$
where $p_i\ge 0$, $k_i$ and $b$ are known for $i=1,\cdots, n$. I'd like to know how to find the solution $\...
- 131
2
votes
1
answer
1k
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Solving a simple Schrödinger equation with Fast Fourier Transforms
While trying to solve a stochastic Gross-Pitaevskii equation I have found a problem that can be tracked down to something buggy occurring in the simplest Schrodinger equation possible:
$$\partial_t \...
- 21
2
votes
1
answer
221
views
Discrete curve-shortening flow – numerical implementation
I need to investigate the properties of open curves which evolve according to the standard curve-shortening flow (Wikipedia link), but with fixed extremes as boundaries (si it should converge to the ...
- 31
2
votes
1
answer
243
views
Simulating Fisher Equation (FKPP)
I'm researching about microbial growth (on 2D). I think that a microbial population can be modeled by the Fisher equation (any other suggestion is welcomed). My doubt is about how can I solve ...
- 121
2
votes
1
answer
194
views
Finite Element Method on a single triangular element
Consider the Laplace equation on a single triangular domain with a Dirichlet condition on two of the sides and a Neumann condition on the remaining side. I am using a linear element ... $\mathbb{P}_1$ ...
- 171
2
votes
0
answers
68
views
Are there spectral Galerkin methods for PDE of the form $\partial_tu=\nabla\cdot f(\nabla u)\nabla u$?
Question is in the title. The nonlinearity due to the term $f(\nabla u)$ makes it difficult to directly apply the spectral Galerkin method as it can be done for PDE of the form $\partial_tu=\nabla\...
- 167
2
votes
0
answers
30
views
Free programs suggestions to simulate parabolic EDPs
I'm interested in learning how to computationally simulate the behavior of parabolic partial differential equations, but I don't know where to start, what are the best free programs to use and where ...
- 677
2
votes
0
answers
395
views
Numerical Method Simulation for 2D Advection Diffusion Equation on Python [closed]
Here it is an Advection-Diffusion equation in 2D:
$$
\frac{\partial C}{\partial t}+U \frac{\partial C}{\partial x}+V \frac{\partial C}{\partial y}=D\left(\frac{\partial^2 C}{\partial x^2}+\frac{\...
2
votes
0
answers
141
views
Approximating solutions to Monge-Ampere from optimal transport plans
I am interested in finding numerical solutions to a Monge-Ampere type equation for applications in physics. Due to the close connection between Monge-Ampere and optimal transport and the well ...
- 956
2
votes
0
answers
88
views
Kolmogorov $\epsilon$-entropy, $n$-width, and $\epsilon$-capacity and applications
What is the relationship between Kolmogorov $\epsilon$-entropy, Kolmogorov $n$-width, and Kolmogorov $\epsilon$-capacity of a set $M$ in a metric space $X$? (The $\epsilon$-capacity here is the ...
- 131
2
votes
0
answers
106
views
Derivation of the Cahn-Hilliard PDE from the point of view of finite difference methods
Consider the Cahn-Hilliard equation
$$\frac{\partial c}{\partial t} = \nabla^2(f(c)-\varepsilon^2 \nabla^2 c)$$
defined on your favorite domain. I'm looking for a literature reference that formally ...
- 53
2
votes
0
answers
136
views
Gradient $L^\infty$-estimate for heat equation with homogeneous Dirichlet boundary condition
$\Omega\subset \mathbb{R}^N$ is a bounded smooth domain. Consider the homogeneous heat equation with zero boundary condition in $\Omega$
\begin{cases}
\partial_t u-\Delta u=0 \quad(x,t)\in \Omega\...
- 121