Questions tagged [numerical-analysis-of-pde]
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99
questions
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What the scientists in numerical solutions of PDE are concerned recently? [closed]
I'm a new one in the numerical solution of PDE, mainly interest in NS equations, Convection-Diffusion equations and I know a lot of people are solving equations by ML, but I really don't like this ...
0
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0
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31
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Convergence of numerical scheme for HJB equation
Convergence of numerical scheme for HJB equation has been widely studied, the key paper is the Barles's one. Essentially, the convergence is guarenteed if the scheme is:
Consistent
Stable
Monotony
...
0
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0
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132
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Relative bounds for vorticity
Write the vorticity equation as
\begin{equation}\label{Eq20}
\begin{split}
\dfrac{\partial}{\partial t} v_i & = \biggl[|\textbf{v}|~|\nabla u_i|\cos(\beta_i)- |\textbf{u}|~|\nabla v_i|\cos(\...
0
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0
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49
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How to solve with FEM a semilinear elliptic equation?
I searched in many books regarding FEM how to solve semilinear elliptic equation, but I did not find too many things. They mostly treat linear and simple problems. For example in P.Ciarlet-The finite ...
0
votes
1
answer
137
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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. ...
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77
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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 \...
3
votes
1
answer
230
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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 ...
2
votes
0
answers
64
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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\...
1
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0
answers
292
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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 $(u,v)\in [0,a]^2$.
$$
\begin{cases}
\tau_u&=F\left( \gamma,\gamma_u,\gamma_v,\...
3
votes
2
answers
409
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Inconsistency in determinability of the solution of a linear first order PDE
Consider the following differential equation:
$$\frac{\partial u(x,t)}{\partial t} = - \frac{\partial u(x,t)}{\partial x} + u(x,t) \label{1}\tag{1}$$
with $u(x,0)=f(x)$. The solution of \eqref{1}, ...
7
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2
answers
570
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What are dissipative PDEs?
I often come across the term dissipative (partial) differential equation in mathematical articles, especially in the context of hypocoercivity and entropy methods. I now have an intuitive idea of ...
1
vote
0
answers
35
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Studying the evolution of laplacian in NS equation
The Navier-Stokes equation in $\mathbb{R}^3$ subjected to no gravitational forces are provided by:
\begin{equation}\label{Eq1}
\dfrac{\partial }{\partial t} \textbf{u} + \left(\textbf{u}\cdot \nabla \...
1
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0
answers
51
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Error estimates for inhomogeneous semidiscrete PDE
I have the following semidiscrete problem on a meshed domain $U_h$. Let
$V_h$ be linear finite elements on $U_h$, $V_{h0}\subset V_h$ have zero trace on $\partial \Omega_h$, and
$V_{h\partial}$ be ...
3
votes
0
answers
69
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Confusion with implementation of PDE constraint Bayesiain inverse problem
Consider a PDE,
$$\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(...
1
vote
1
answer
105
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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 ...
2
votes
0
answers
27
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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 ...
2
votes
0
answers
246
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Numerical Method Simulation for 2D Advection Diffusion Equation on Python [closed]
Here it is an Advection-Diffusion equation in 2D:
$$
\frac{\partial C}{\partial t}+U \frac{\partial C}{\partial x}+V \frac{\partial C}{\partial y}=D\left(\frac{\partial^2 C}{\partial x^2}+\frac{\...
0
votes
0
answers
39
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Rigorous definition of space and time order of accuracy of numerical PDEs
Suppose that we are solving numerically a PDE (with a numerical scheme like this one) which involves space $x$ and time $t.$ It is a commonly seen expression in the literature that "the method ...
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27
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Generating a proper finite difference scheme
I have recently started studying the finite difference schemes for numerical analysis. While I can now calculate difference schemes fairly easily for simple equations, I've recently come across a ...
2
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0
answers
108
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Approximating solutions to Monge-Ampere from optimal transport plans
I am interested in finding numerical solutions to a Monge-Ampere type equation for applications in physics. Due to the close connection between Monge-Ampere and optimal transport and the well ...
3
votes
1
answer
723
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Navier-Stokes equations and machine learning
I am looking for a reference explaining how to solve the Navier-Stokes equations numerically using machine learning algorithms .
Thank you in advance for your help .
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66
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Computing and isotopy of curves in $\mathbb{R}^3$
Imagine a piece of string in the ocean moving gently with the currents; the string bends but does not change its length.
The (stationary) string can be modelled by a unit speed curve:
$$[0,1] \...
1
vote
1
answer
182
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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
$$\dfrac{\partial}{\partial t}\...
4
votes
2
answers
552
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How to compute $\sin(\frac{d}{dx})f(x)$?
Assuming $f(x)=e^{-x^2}$ for $x$ in $[-10,10]$, I have tried the following:
Fourier transform $\mathcal{F}$: $\frac{d}{dx}$ can be diagonalized as $\mathcal{F}^{-1} i\omega \mathcal{F}$. Therefore, $\...
1
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0
answers
73
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Highy non-linear PDE involving directional derivative
Let the convolution of two function $f$ and $g$ be defined over $\mathbb{R}^3\times [0,\infty)$ as followed
\begin{equation}\label{ConvoDef}
\left(f*g\right)\circ(\textbf{x},t) = \int_{0}^{t}{\int_{\...
3
votes
2
answers
355
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Questions for the non-linear PDE $2u_t=\log(-u_{xx})$
Consider the PDE as follows :
$$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} \...
2
votes
0
answers
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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 ...
2
votes
2
answers
269
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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 ...
1
vote
0
answers
52
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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 ...
2
votes
0
answers
86
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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 ...
0
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1
answer
105
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FEM based solution to parabolic problem
Consider the problem
$$
\begin{cases}
u_t - \Delta u = 0 &\text{ on } \Omega\times (0,T)\\u=0 &\text{ on } \partial \Omega\times (0,T) \\ u(x,0)=g(x) &\text{ on } \Omega
\end{cases}
$$
...
2
votes
0
answers
115
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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\...
3
votes
0
answers
61
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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 ...
2
votes
1
answer
264
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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 ...
1
vote
0
answers
40
views
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 ...
1
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0
answers
56
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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 ...
1
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0
answers
130
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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 = ...
1
vote
1
answer
90
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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 ...
3
votes
0
answers
60
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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} \...
2
votes
0
answers
69
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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 ...
5
votes
1
answer
106
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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, ...
3
votes
0
answers
109
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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{...
3
votes
0
answers
72
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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 ...
6
votes
1
answer
532
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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 ...
0
votes
1
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264
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Analytical testcase for 2D/3D anisotropic Diffusion (Heat Kernel)
I want to verify and compare different Discretizations of the anisotropic diffusion equation in 2D / 3D image of my testsetting.
I want to verify and compare different Discretizations of the ...
1
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0
answers
86
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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 $$\sup_{t\in[0,\:\tau]}\left\|v(t,x)-v(t,y)...
3
votes
1
answer
273
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Conditions on the velocity ensuring that a flow moves points along the boundary of a manifold
Let
$\tau>0$;
$d\in\mathbb N$;
$v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ be Lipschitz continuous in the second argument uniformly with respect to the first with $v(\;\cdot\;,x)\in C^0([0,\tau],\...
0
votes
0
answers
134
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Non linear second order PDE involving max operator (Dynamic Programming)
I'm trying to solve the following Dynamic Programming equation in continuous time ($dt \rightarrow 0$)
$$ 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\} - \...
1
vote
0
answers
74
views
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)...
3
votes
1
answer
376
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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, ...