Questions tagged [numerical-analysis-of-pde]

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Can we study concavity of vorticity equation?

The vorticity equation is well known given as \begin{equation}\label{Eq1} \dfrac{\partial}{\partial t}\textbf{v} + (\textbf{u}\cdot \nabla )\textbf{v} - (\textbf{v}\cdot \nabla )\textbf{u} = \nu \...
ROY BURSON's user avatar
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Stokes equation and Helmholtz decomposition

I apologize in advance for the long question, but it involves some work I been thinking about and would like help with. The Navier-Stokes equation in $\mathbb{R}^3$ subjected to no gravitational ...
ROY BURSON's user avatar
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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\...
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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 ...
miao zhengwu's user avatar
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Solutions of a Gauss–Codazzi-like system of nonlinear PDEs

Consider the following system of PDEs for the dependent variables $\tau=\tau(u,v)$ and $\gamma=\gamma(u,v)$, with $(u,v)\in [0,a]^2$. $$ \begin{cases} \tau_u&=F\left( \gamma,\gamma_u,\gamma_v,\...
Daniel Castro's user avatar
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2 answers
382 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}, ...
Mirar's user avatar
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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 ​​...
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Studying the evolution of laplacian in NS equation

The Navier-Stokes equation in $\mathbb{R}^3$ subjected to no gravitational forces are provided by: \begin{equation}\label{Eq1} \dfrac{\partial }{\partial t} \textbf{u} + \left(\textbf{u}\cdot \nabla \...
MrPie 's user avatar
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Where can I find the paper by Tappert and Hardin on split-step Fourier transform method?

The split-step method is a numerical method that can be used to solve a nonlinear PDE (https://en.wikipedia.org/wiki/Split-step_method). Even Wikipedia does not refer to the original authors (F.D. ...
Redsbefall's user avatar
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Error estimates for inhomogeneous semidiscrete PDE

I have the following semidiscrete problem on a meshed domain $U_h$. Let $V_h$ be linear finite elements on $U_h$, $V_{h0}\subset V_h$ have zero trace on $\partial \Omega_h$, and $V_{h\partial}$ be ...
Lilla's user avatar
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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(...
Jarwin's user avatar
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How to numerically solve differential equations involving sines, cosines and inverses of the unknown function? [closed]

Crossposted at SciComp SE I'm very new to finite difference method and I am just introduced to methods of solving differential equation using finite difference method via sparse matrix method. I find ...
Hari Sam's user avatar
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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 ...
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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{\...
Edric Jonathan's user avatar
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Rigorous definition of space and time order of accuracy of numerical PDEs

Suppose that we are solving numerically a PDE (with a numerical scheme like this one) which involves space $x$ and time $t.$ It is a commonly seen expression in the literature that "the method ...
affine_scheme's user avatar
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Generating a proper finite difference scheme

I have recently started studying the finite difference schemes for numerical analysis. While I can now calculate difference schemes fairly easily for simple equations, I've recently come across a ...
Syed Ali Mohsin Bukhari's user avatar
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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 ...
Yly's user avatar
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Computing and isotopy of curves in $\mathbb{R}^3$

Imagine a piece of string in the ocean moving gently with the currents; the string bends but does not change its length. The (stationary) string can be modelled by a unit speed curve: $$[0,1] \...
sitiposit's user avatar
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PDE involving curl

Let $G:\mathbb{R}^3\rightarrow\mathbb{R}^3$ be smooth vector field over $\mathbb{R}^3$. For which vector fields $F:\mathbb{R}^3\rightarrow\mathbb{R}^3$ does the PDE $$\dfrac{\partial}{\partial t}\...
MrPie 's user avatar
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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, $\...
Mirar's user avatar
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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 ...
Hiro's user avatar
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Highy non-linear PDE involving directional derivative

Let the convolution of two function $f$ and $g$ be defined over $\mathbb{R}^3\times [0,\infty)$ as followed \begin{equation}\label{ConvoDef} \left(f*g\right)\circ(\textbf{x},t) = \int_{0}^{t}{\int_{\...
MrPie 's user avatar
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$H^s$ 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 ...
Chushamm's user avatar
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Flux that can be represented by low and high resolution schemes

In the wiki page of Flux limiter, it writes: If these edge fluxes can be represented by low and high resolution schemes, then a flux limiter can switch between these schemes depending upon the ...
Ho-Oh's user avatar
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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 ...
ithmath's user avatar
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FEM based solution to parabolic problem

Consider the problem $$ \begin{cases} u_t - \Delta u = 0 &\text{ on } \Omega\times (0,T)\\u=0 &\text{ on } \partial \Omega\times (0,T) \\ u(x,0)=g(x) &\text{ on } \Omega \end{cases} $$ ...
user481867's user avatar
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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\...
Tibeku's user avatar
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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 ...
mohammad fazli's user avatar
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P1-finite element as convolution of P0-finite element

For a vector $u\in\mathbf{R}^N$ let's denote $\pi_N(u)$ the unique piecwise linear and $1$-periodic function matching the components of $u$ on the discretization $x_k = \frac{k}{N}$ of the unit ...
Ayman Moussa's user avatar
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How does a computer program recognize shocks given data of a solution to a conservation law?

Conservation laws are PDEs of the form $u_t +j_x=0.$ A discontinuous solution (for $u$ and $j$) to an equation like this can be easily found. Let's suppose that we are working with a piecewise ...
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Semilinear PDE - BSDE presentation via Feynman Kac formula

For a semilinear PDE, we usually have this FBSDE representation: $\mathcal{X}_t=\mathcal{X}_0+\int^t_0 \mu (s,\mathcal{X}_s)\, ds\, +\int_0^t \sigma (s,\mathcal{X}_s)dW_s,\quad 0\leq t\leq T, \\ Y_t = ...
freshst4r's user avatar
1 vote
1 answer
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Typo in error a-priori estimate in a discontinuous Galerkin paper?

I'm looking at this famous paper which is available in the link below: Franco Brezzi, LD Marini, Endre Süli, Discontinuous Galerkin methods for first-order hyperbolic problems, Mathematical Models ...
bobinthebox's user avatar
3 votes
0 answers
60 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} \...
Onil90's user avatar
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2 votes
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Dense matrix vs sparse matrix, when they have same number of nonzero elements

I came across a new way in the literature to solve PDE problems numerically, which is called 'Patch Reconstruction'. One example paper is: Li, R., Sun, Z., Yang, F., & Yang, Z. (2019). A finite ...
陳Keefe's user avatar
3 votes
2 answers
346 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} \...
GJC20's user avatar
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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 ...
Ken Hung's user avatar
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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{...
Alex's user avatar
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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 ...
Sébastien Loisel's user avatar
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Shape derivative at manifold $M$ in direction $v$ is equal to the shape derivative at $\partial M$ in drection $\langle v,n\rangle n$

Let $\tau>0$ and $d\in\mathbb N$. Definiton 1$\:\:\:$If $v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ with $v(\;\cdot\;,x)\in C^0([0,\tau],\mathbb R^d)$ and $$\sup_{t\in[0,\:\tau]}\left\|v(t,x)-v(t,y)...
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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],\...
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Show that the support of the shape gradient $\nabla\mathcal F(\Omega)$ is contained in $\overline\Omega$

Let $E$ be a $\mathbb R$-Banach space, $\Theta\subseteq C^{0,\:1}(E,E)$ be a $\mathbb R$-Banach space and $(T^{(\theta)}_t)_{t\ge0}$ denote the $C^1$-diffeomorphism from $E$ onto $E$ with $$T^{(\theta)...
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Non linear second order PDE involving max operator (Dynamic Programming)

I'm trying to solve the following Dynamic Programming equation in continuous time ($dt \rightarrow 0$) $$ v(x,t) = \max\Big\{|x|\,,\,v(x,t)+dt\Big(v_t(x,t)+\frac{1}{2(t+1)}v_{xx}(x,t)\Big) \Big\} - \...
J. R. C.'s user avatar
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3 votes
1 answer
375 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 ...
0xbadf00d's user avatar
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1 vote
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Von Neumann analysis on a finite difference hyperbolic scheme

I am doing a Von Neumann analysis on a staggered finite difference scheme (for Maxwell's Equations). The finite difference scheme is: $$ \mathbf{u}_v|^{n+2}_{i,j} - \mathbf{u}_v|^{n}_{i,j} = - A \frac{...
Barros's user avatar
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Differentiation under the integral sign for a $L^1$-valued function (shape derivative)

Let $d\in\mathbb N$; $U\subseteq\mathbb R^d$ be open and $$\mathcal A:=\{\Omega\subseteq U:\Omega\text{ is bounded and open and }\partial\Omega\text{ is of class }C^{0,\:1}\};$$ $E:=\bigcup_{\Omega\...
0xbadf00d's user avatar
  • 131
3 votes
1 answer
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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, ...
0xbadf00d's user avatar
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3 votes
1 answer
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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 ...
user142929's user avatar
1 vote
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Choice of finite element spaces in plasticity

I am planning to run numerical simulations in metal elastoplasticity (von-Mises yield condition with and without isotropic hardening). However, I am completely new to this subject and I am unsure ...
phlegmax's user avatar
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2 votes
1 answer
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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 ...
rndm_ecn's user avatar
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68 views

Fokker-Planck equation with zero diffusion coefficients on the boundaries

I am currently working on a Masters project in which I will be looking at numerical methods for a specific PDE. The PDE that I will be considering is a Fokker-Planck equation on a unit hyper-cube. ...
Jandré Snyman's user avatar